A Class of Nested Iteration Schemes for Generalized Coupled Sylvester Matrix Equation

Document Type : Research Paper


1 Department of Applied Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran

2 Department of Applied Mathematics, Faculty of Mathematics & Computer Sciences, Shahid Bahonar University of Kerman

3 Department of Applied Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran.



Global Krylov subspace methods are the most efficient and robust methods to solve generalized coupled Sylvester matrix equation. In this paper, we propose the nested splitting conjugate gradient process for solving this equation. This method has inner and outer iterations, which employs the generalized conjugate gradient method as an inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergence and symmetric positive definite splitting of the coefficient matrices. Convergence properties of this method are investigated. Finally, the effectiveness of the nested splitting conjugate gradient method is explained by some numerical examples.


[1] O. Axelsson, Z.-Z. Bai and S.-X. Qiu, A class of nested iteration schemes for linear systems with a
      coefficient matrix with a dominant positive definite symmetric part, Numer. Algorithms., 21(35) (2004),    
[2] R. Boisvert, R. Pozo, K. Remington, B. Miller and R. Lipman, Matrix Market, National Institute of Standards 
      and Technology, http:// math.nist.gov/ matrixMarket/, 1996.
[3] A. Bouhamidi and K. Jbilou, A note on the numerical approximate solutions for generalized Sylvester    
     matrix equations with applications, Appl. Math. Comput., 206 (2008), 687-694.
[4] M. Dehghan and M. Hajarian, An iterative method for solving the generalized coupled Sylvester matrix 
      equations over generalized bisymmetric matrices, Appl. Math. Model., 34 (2010), 639-588.
[5] R.A. Horn and C.R. Jahnson, Matrix Analysis, Cambridge University Press, United Kingdom, second edition, 
[6] K. Jbilou, A. Messaoudi and H. Sadok, Global FOM and GMRES algorithms for matrix equations, Appl.
      Numer. Math., 31 (1999), 49-63.
[7] K. Jbilou and A.J. Riquet, Projection methods for large Lyapunov matrix equations, Linear Algebra Appl.,
       415 (2006), 344-358.
[8] Y.F. Ke and C.F. Ma, A preconditioned nested splitting conjugate gradient iterative method for the large   
      sparse generalized Sylvester equation, Comput. Math. Appl., 68(10) (2014), 1409-1420.
[9] C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadelphia, 1995.
[10] M. Khorsand Zak and F. Toutounian, An iterative method for solving the continuous Sylvester equation by 
       emphasizing on the skew-Hermitian parts of the coeffcient matrices, Int. J. Comput. Math., 94 (2017), 
[11] M. Khorsand Zak and F. Toutounian, Nested splitting CG-like iterative method for solving the continuous 
       Sylvester equation and preconditioning, Adv. Comput. Math., 40(4) (2014), 865-880.
[12] M. Khorsand Zak and F. Toutounian, Nested splitting conjugate gradient method for matrix equation  
       AXB  = C and preconditioning, Comput. Math. Appl., 66(3) (2013), 269-278.
[13] F. Panjeh Ali Beik and D. Khojasteh Salkuyeh, On the global Krylov subspace methods for solving general
       coupled matrix equations, Comput. Math. Appl., 62 (2011), 4605-4613.
[14] F. Panjeh Ali Beik and D. Khojasteh Salkuyeh, The coupled Sylvester- transpose matrix equations over 
       generalized centro-symmetric matrices, International J. Comput. Math., 90(7) (2013), 1546-1566.
[15] J.J. Zhang, A note on the iterative solutions of general coupled matrix equation, Appl. Math. Comput., 217 
       (2011), 8386-9380.
[16] B. Zhou and G.R. Duan, On the generalized Sylvester mapping and matrix equation, Systems Control 
       Lett., 57(3) (2008), 200-2008.