A new projection method for solving large Sylvester equations

In this paper, we propose a new projection method based on global Arnoldi algorithm for solving large Sylvester matrix equations AX+XB+CD^T=0 and the large generalized Sylvester matrix equations of the form AXB+X+CD^T=0. We show how to extract low-rank approximate solutions to Sylvester matrix equations and generalized Sylvester matrix equations. Some theoretical results are given. Numerical tests report the effectiveness of these methods.

[1]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[2]  K. Jbilou,et al.  Projection methods for large Lyapunov matrix equations , 2006 .

[3]  D. Bernstein,et al.  The optimal projection equations for fixed-order dynamic compensation , 1984 .

[4]  G. Golub,et al.  A Hessenberg-Schur method for the problem AX + XB= C , 1979 .

[5]  Jacob K. White,et al.  Low Rank Solution of Lyapunov Equations , 2002, SIAM J. Matrix Anal. Appl..

[6]  B. Datta.  Numerical methods for linear control systems : design and analysis , 2004 .

[7]  Yimin Wei,et al.  Krylov subspace methods for the generalized Sylvester equation , 2006, Appl. Math. Comput..

[8]  I. Jaimoukha,et al.  Krylov subspace methods for solving large Lyapunov equations , 1994 .

[9]  Richard H. Bartels,et al.  Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4] , 1972, Commun. ACM.

[10]  J. Hearon,et al.  Nonsingular solutions of TA−BT=C , 1977 .

[11]  Lothar Reichel,et al.  On the solution of large Sylvester-observer equations , 2001, Numer. Linear Algebra Appl..

[12]  Khalide Jbilou,et al.  Block Krylov Subspace Methods for Solving Large Sylvester Equations , 2002, Numerical Algorithms.

[13]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

[14]  L. Reichel,et al.  Krylov-subspace methods for the Sylvester equation , 1992 .

[15]  Shankar P. Bhattacharyya,et al.  Controllability, observability and the solution of AX - XB = C , 1981 .

[16]  Anders Lindquist,et al.  Computational and combinatorial methods in systems theory , 1986 .

[17]  H. Sadok,et al.  Global FOM and GMRES algorithms for matrix equations , 1999 .

[18]  Miloud Sadkane,et al.  A Convergence Analysis of Gmres and Fom Methods for Sylvester Equations , 2002, Numerical Algorithms.

[19]  A. Laub,et al.  Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms , 1987 .