**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30831

##### The BGMRES Method for Generalized Sylvester Matrix Equation AXB − X = C and Preconditioning

**Authors:**
Azita Tajaddini,
Ramleh Shamsi

**Abstract:**

**Keywords:**
Linear matrix equation,
Block GMRES,
matrix Krylov
subspace,
polynomial preconditioner

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1131245

**References:**

[1] L. Bao, Y. Lin and Y. Wei, ”Krylov subspace methods for the generalized Sylvester equation”, Appl. Mathem. Comput. vol. 175, 2006, pp. 557–573.

[2] D. S. Bernstein and W. M. Haddad, ”LQG control with a Hinf performance bound: a Riccati equation approach”, IEEE Trans. Automat. Control, vol. AC-34, 1989, pp. 293-305.

[3] A. Bouhamidi and K. Jbilou, ”A note on the numerical approximate solutions for generalized sylvester matrix equations with applications”, Appl. Math. Comput., vol. 206, 2008, pp. 687–694.

[4] J. W. Demmel, Applied numerical linear algebra; SIAM, 1997.

[5] Z. Gajic and M. T. J. Qureshi, Lyapunov Matrix Equation in System Stability and Control; Dover, Mineola, NY, 2008.

[6] G. H. Golub, S. Nash and C. F. Vanloan , ”A Hessenberg- Schur method for problem AX + XB = C”, IEEE Trans. Automat. Contr., vol. 24, 1979, pp. 909–913.

[7] G. H. Golub, S. Nash and C. F. Vanloan, Matrix Computations, Jhons Hoplins U. P., Baltimore, 3th edn, 1996.

[8] K. Jbilou, A. Messaoudi and A. Sadok, ”Global FOM and GMRES algorithms for matrix equation”, Appl. Numer. Math., vol. 31, 1999, pp. 49–63.

[9] D. Khojasteh Salkuyeh and F. Toutounian, ”New approaches for solving large sparse Sylvester equations”, Appl. Math. Comput., vol. 173, no. 1, 2006, pp. 9–18.

[10] T. Li, P. Chang-Yi Weng,E. King- Wash Chu and W. W. Lin, ”Large-Scale Stein and Lyapunov equations, Smith methods and applications”, Numer. Algor, vol. 63, 2013, pp. 727–752.

[11] M. Mohseni Moghadam, A. Rivaz, A. Tajaddini and F. Saberi Mouvahed, ”Convergence analysis of the global FOM and GMRES methods for solving matrix equations AXB = C with spd coefficients”, Bull. Iranian Math. Soc, vol. 41, no. 4, 2015, pp. 981–1001.

[12] B. C. Moore, ”Principal component analysis in linear systems: controllability, observability, and model reduction”, IEEE Trans. Automat. Contr., vol. AC-26, 1981, pp. 17-32.

[13] F. Panjeh Ali Beik, ”Note to the global GMRES for solving the matrix equation AXB = C”, Int. J. Eng. Nat. Sci., vol. 5, no. 2, 2003, pp. 101–105.

[14] Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, 2003.

[15] J. Saak, Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction, Dr. Rer. Nat. dissertation, Chemnitz University of Technology, Germany, 2009.

[16] M. Sadkane, ”A low rank Krylov squared Smith method for large-scale discrete-time Lyapunov equations”, Linear Algebra Appl., vol. 436, 2012, pp. 2807–2827.

[17] J. L. Salle and Lefschetz, Stability by Lyapunovs Direct Method with Applications, Academic press, 1961.

[18] M. G. Sanfonov and R. Y. Chiang, A Schur method for balanced model reduction, Proc. Amer. Control Conf., Atlanta, GA, 1988.

[19] K. Yi-Fen and M. Chang- Feng, preconditioned nested splitting conjugate gradient iterative method for the large sparse generalized Sylvester equation, Computers and Mathematics with Applications, vol. 68, no. 10, 2014, pp. 1409–1420.