An iterative scheme for a class of generalized Sylvester matrix equations

Document Type : Original Article


1 Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Iran

2 Faculty of Basic Sciences, Shahid Sattari Aeronautical University of Sciences and Technology, South Mehrabad, Tehran, Iran


In this study based on the accelerated over relaxation (AOR) method we make an iterative scheme for solving generalized Lyapunov matrix equation
$$ {A}{X}+{X}{B}+\sum_{j=1}^{m}{N}_j{X}{M}_j={C},$$
over complex or real matrices. Then we analyze the convergence of the new iterative method in detail. There have been discussions for the calculation of optimal parameters. Finally a numerical example is given to demonstrate the capability of the new method.


Main Subjects

  1. Ali, I. Khan, A. Ali, and A. Mohamed, Two new generalized iteration methods for solving absolute value equations using M-matrix, AIMS Math., 7 (2022), pp. 8176–8187.
  2. Ali, K. Pan, and A. Ali, Two generalized successive overrelaxation methods for solving absolute value equations, Math. Theory Appl., 40 (2020), pp. 44–55.
  3. Axelsson, Iterative Solution Methods, Cambridge University Press, 1994.
  4. -Z. Bai, On Hermitian and skew-Hermitian splitting iteration methods for continuous Sylvester equations, J. Comput. Math., 29 (2011), pp. 185–198.
  5. -Z. Bai, X.-X. Guo, and 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), pp. 655–674.
  6. -Z. Bai, B. N. Parlett, and Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 102 (2005), pp. 1–38.
  7. H. Bartels and G. W. Stewart, Algorithm 432: Solution of the matrix equation AX + XB = C, Commun. ACM, 15 (1972), pp. 820–826.
  8. P. A. Beik, M. Najafi-Kalyani, and L. Reichel, Iterative Tikhonov regularization of tensor equations based on the Arnoldi process and some of its generalizations, Appl. Numer. Math., 151 (2020), pp. 425–447.
  9. Benner, Factorized solution of sylvester equations with applications in control, in Proceedings of the 16th International Symposium on Mathematical Theory of Network and Systems, B. D. Moor, B. Motmans, J. Willems, P. V. Dooren, and V. Blondel, eds., Leuven, Belgium, 5–9 July 2004.
  10. , Large-scale matrix equations of special type, Numer. Linear Algebra Appl., 15 (2008), pp. 747–754.
  11. Benner and T. Breiten, Low rank methods for a class of generalized Lyapunov equations and related issues, Numer. Math., 124 (2013), pp. 441–470.
  12. Bertaccini, Efficient preconditioning for sequences of parametric complex symmetric linear systems, Electron. Trans. Numer. Anal., 18 (2004), pp. 49–64.
  13. Bouhamidi and K. Jbilou, Sylvester Tikhonov-regularization methods in image restoration, J. Comput. Appl. Math., 206 (2007), pp. 86–98.
  14. , A note on the numerical approximate solutions for generalized Sylvester matrix equations with applications, Appl. Math. Comput., 206 (2008), pp. 687–694.
  15. Damm, Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations, Numer. Linear Algebra Appl., 15 (2008), pp. 853–871.
  16. Dehghan and M. Hajarian, An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation, Appl. Math. Comput., 202 (2008), pp. 571–588.
  17. , An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices, Appl. Math. Model., 34 (2010), pp. 639–654.
  18. , Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations, Appl. Math. Model., 35 (2011), pp. 3285–3300.
  19. Dehghan and A. Shirilord, A generalized modified Hermitian and skew-Hermitian splitting (GMHSS) method for solving complex Sylvester matrix equation, Appl. Math. Comput., 348 (2019), pp. 632–651.
  20. , A new approximation algorithm for solving generalized Lyapunov matrix equations, J. Comput. Appl. Math., 404 (2022), Paper No. 113898, 26.
  21. S. A. Dilip and H. K. Pillai, Characterization of solutions of non-symmetric algebraic Riccati equations, Linear Algebra Appl., 507 (2016), pp. 356–372.
  22. Ding, P. X. Liu, and J. Ding, Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Appl. Math. Comput., 197 (2008), pp. 41–50.
  23. Ding and H. Zhang, Gradient-based iterative algorithm for a class of the coupled matrix equations related to control systems, IET Control Theory Appl., 8 (2014), pp. 1588–1595.
  24. Ding, Y. Liu, and F. Ding, Iterative solutions to matrix equations of the form AiXBi = Fi, Comput. Math. Appl., 59 (2010), pp. 3500–3507.
  25. -Y. Fan, P. C.-Y. Weng, and E. K.-W. Chu, Numerical solution to generalized Lyapunov/Stein and rational Riccati equations in stochastic control, Numer. Algorithms, 71 (2016), pp. 245–272.
  26. Hadjidimos, Accelerated overrelaxation method, Math. Comp., 32 (1978), pp. 149–157.
  27. -H. He, Q.-W. Wang, and Y. Zhang, A simultaneous decomposition for seven matrices with applications, J. Comput. Appl. Math., 349 (2019), pp. 93–113.
  28. B. Iantovics and F. F. Nichita, On the colored and the set–theoretical Yang–Baxter equations, Axioms, 10 (2021), p. 146.
  29. Karamali, A. Shirilord, and M. Dehghan, On the CRI method for solving Sylvester equation with complex symmetric positive semi-definite coefficient matrices, Filomat, 35 (2021), pp. 3071–3090.
  30. Ke and C. Ma, The alternating direction methods for solving the Sylvester-type matrix equation AXB + CXD = E, J. Comput. Math., 35 (2017), pp. 620–641.
  31. Li, A.-L. Yang, and Y.-J. Wu, Lopsided PMHSS iteration method for a class of complex symmetric linear systems, Numer. Algorithms, 66 (2014), pp. 555–568.
  32. Mukaidani, H. Xu, and K. Mizukami, Numerical Algorithm for Solving Cross-Coupled Algebraic Riccati Equations of Singularly Perturbed Systems, Birkh¨auser Boston, Boston, MA, 2005, pp. 545–570.
  33. A. Ramadan, T. S. El-Danaf, and A. M. E. Bayoumi, A relaxed gradient based algorithm for solving extended Sylvester-conjugate matrix equations, Asian J. Control, 16 (2014), pp. 1334–1341.
  34. A. Smith, Matrix equation XA + BX = C, SIAM J. Appl. Math., 16 (1968), pp. 198–201.
  35. -W. Wang, The general solution to a system of real quaternion matrix equations, Comput. Math. Appl., 49 (2005), pp. 665–675.
  36. -W. Wang, Z.-H. He, and Y. Zhang, Constrained two-sided coupled Sylvester-type quaternion matrix equations, Automatica J. IFAC, 101 (2019), pp. 207–213.
  37. Xie, N. Huang, and C. Ma, Iterative method to solve the generalized coupled Sylvester-transpose linear matrix equations over reflexive or anti-reflexive matrix, Comput. Math. Appl., 67 (2014), pp. 2071–2084.
  38. Zhang and H. Kang, The generalized modified Hermitian and skew-hermitian splitting method for the generalized lyapunov equation, Int. J. Control Autom. Syst., 19 (2021), pp. 339–349.
  39. Zhou and G.-R. Duan, Solutions to generalized Sylvester matrix equation by Schur decomposition, Internat. J. Systems Sci., 38 (2007), pp. 369–375.
  40. , On the generalized Sylvester mapping and matrix equations, Systems Control Lett., 57 (2008), pp. 200– 208.