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

Authors: Azita Tajaddini, Ramleh Shamsi

Abstract:

In this paper, we present the block generalized minimal residual (BGMRES) method in order to solve the generalized Sylvester matrix equation. However, this method may not be converged in some problems. We construct a polynomial preconditioner based on BGMRES which shows why polynomial preconditioner is superior to some block solvers. Finally, numerical experiments report the effectiveness of this method.

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

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

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[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.