AUT Journal of Mathematics and Computing

AUT Journal of Mathematics and Computing

Construction of an iterative method for solving a class of complex symmetric generalized Lyapunov matrix equation and application to Helmholtz equation

Document Type : Original Article

Authors
Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology (Tehran polytechnic), No. 424, Hafez Ave., 15914, Tehran, Iran
Abstract
The Lyapunov matrix equations occur in many branches of control theory, such as stability analysis and optimal control. In this work, we introduce a novel iterative approach to address the generalized Lyapunov matrix equation within the framework of complex matrices. At each iteration, the procedure involves solving two conventional Lyapunov equations with real-valued coefficient matrices. The scheme incorporates two positive parameters, for which we establish sufficient conditions to guarantee the convergence of the method under certain assumptions. Then we solve the Lyapunov equation arising by applying a finite difference procedure to Helmholtz equation by proposed method.
Keywords
Subjects

[1]     Z.-Z. Bai, On Hermitian and skew-Hermitian splitting iteration methods for continuous Sylvester equations, J. Comput. Math., 29 (2011), pp. 185–198.
[2]     Z.-Z. Bai, G. H. Golub, and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), pp. 603–626.
[3]     Z. Baralak, M. Dehghan, F. Fakhar-Izadi, and M. Abbaszadeh, A local discontinuous galerkin spectral element method for high-frequency wave propagation in computational acoustics, Journal of Computational Physics, (2025), p. 114431.
[4]     F. 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.
[5]     D. Bertaccini, Efficient preconditioning for sequences of parametric complex symmetric linear systems, Electron. Trans. Numer. Anal., 18 (2004), pp. 49–64.
[6]     A. Bouhamidi and K. Jbilou, A note on the numerical approximate solutions for generalized Sylvester matrix equations with applications, Appl. Math. Comput., 206 (2008), pp. 687–694.
[7]     K.-w. E. Chu, Singular value and generalized singular value decompositions and the solution of linear matrix equations, Linear Algebra Appl., 88/89 (1987), pp. 83–98.
[8]     P. d’Alessandro, A. Isidori, and A. Ruberti, Realization and structure theory of bilinear dynamical systems, SIAM J. Control, 12 (1974), pp. 517–535.
[9]     M. Dehghan and M. Hajarian, Efficient iterative method for solving the second-order Sylvester matrix equation EV F2 AV F CV = BW, IET Control Theory Appl., 3 (2009), pp. 1401–1408.
[10]  M. Dehghan and M. Hajarian, On the reflexive solutions of the matrix equation AXB + CY D = E, Bull. Korean Math. Soc., 46 (2009), pp. 511–519.
[11]  , An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices, Appl. Math. Model., 34 (2010), pp. 639–654.
[12]  M. Dehghan, M. Nourian, and M. B. Menhaj, Numerical solution of helmholtz equation by the modified hopfield finite difference techniques, Numerical Methods for Partial Differential Equations: An International Journal, 25 (2009), pp. 637–656.
[13]  M. 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.
[14]  , Approximating optimal parameters for generalized preconditioned Hermitian and skew-Hermitian splitting (GPHSS) method, Comput. Appl. Math., 41 (2022), pp. Paper No. 72, 23.
[15]  Y.-B. Deng, Z.-Z. Bai, and Y.-H. Gao, Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations, Numer. Linear Algebra Appl., 13 (2006), pp. 801–823.
[16]  F. Ding and T. Chen, On iterative solutions of general coupled matrix equations, SIAM J. Control Optim., 44 (2006), pp. 2269–2284.
[17]  F. 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.
[18]  M. Hajarian, Least squares solution of the linear operator equation, J. Optim. Theory Appl., 170 (2016), pp. 205–219.
[19]  Z.-H. He, Q.-W. Wang, and Y. Zhang, A system of quaternary coupled Sylvester-type real quaternion matrix equations, Automatica J. IFAC, 87 (2018), pp. 25–31.
[20]  Y. Ke and C. Ma, An alternating direction method for nonnegative solutions of the matrix equation AX + Y B = C, Comput. Appl. Math., 36 (2017), pp. 359–365.
[21]  D. L. Kleinman, On the stability of linear stochastic systems, IEEE Trans. Automatic Control, AC-14 (1969), pp. 429–430.
[22]  X. 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.
[23]  A. Navarra, P. L. Odell, and D. M. Young, A representation of the general common solution to the matrix equations A1XB1 = C1 and A2XB2 = C2 with applications, Comput. Math. Appl., 41 (2001), pp. 929–
935.
[24]  M. A. Ramadan and T. S. El-Danaf, Solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices, Trans. Inst. Meas. Control., 37 (2015), pp. 291–316.
[25]  M. 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.
[26]  D. K. Salkuyeh and M. Bastani, A new generalization of the Hermitian and skew-Hermitian splitting method for solving the continuous Sylvester equation, Trans. Inst. Meas. Control, 40 (2018), pp. 303–317.
[27]  A. Shirilord and M. Dehghan, Combined real and imaginary parts method for solving generalized Lyapunov matrix equation, Appl. Numer. Math., 181 (2022), pp. 94–109.
[28]  , Single step iterative method for linear system of equations with complex symmetric positive semi-definite coefficient matrices, Appl. Math. Comput., 426 (2022), pp. Paper No. 127111, 17.
[29]  , Iterative method for constrained systems of conjugate transpose matrix equations, Appl. Numer. Math., 198 (2024), pp. 474–507.
[30]  , Stationary Landweber method with momentum acceleration for solving least squares problems, Appl. Math. Lett., 157 (2024), pp. Paper No. 109174, 7.
[31]  , Gradient descent-based parameter-free methods for solving coupled matrix equations and studying an application in dynamical systems, Appl. Numer. Math., 212 (2025), pp. 29–59.
[32]  Q.-W. Wang and Fei-Zhang, The reflexive re-nonnegative definite solution to a quaternion matrix equation, Electron. J. Linear Algebra, 17 (2008), pp. 88–101.
[33]  G. Xu, M. Wei, and D. Zheng, On solutions of matrix equation AXB + CY D = F, Linear Algebra Appl., 279 (1998), pp. 93–109.
[34]  B. Zhou and G.-R. Duan, On the generalized Sylvester mapping and matrix equations, Systems Control Lett., 57 (2008), pp. 200–208.
[35]  B. Zhou and Z.-B. Yan, Solutions to right coprime factorizations and generalized Sylvester matrix equations, Trans. Inst. Meas. Control, 30 (2008), pp. 397–426.