Block Krylov Subspace Methods for Solving Large Sylvester Equations

In the present paper, we propose block Krylov subspace methods for solving the Sylvester matrix equation AX−XB=C. We first consider the case when A is large and B is of small size. We use block Krylov subspace methods such as the block Arnoldi and the block Lanczos algorithms to compute approximations to the solution of the Sylvester matrix equation. When both matrices are large and the right-hand side matrix is of small rank, we will show how to extract low-rank approximations. We give some theoretical results such as perturbation results and bounds of the norm of the error. Numerical experiments will also be given to show the effectiveness of these block methods.

[1]  B. Vital.  Etude de quelques methodes de resolution de problemes lineaires de grande taille sur multiprocesseur , 1990 .

[2]  F. Chatelin.  Valeurs propres de matrices , 1988 .

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

[4]  V. N. Bogaevski,et al.  Matrix Perturbation Theory , 1991 .

[5]  V. Simoncini,et al.  On the numerical solution ofAX −XB =C , 1996 .

[6]  R. Freund,et al.  A block QMR algorithm for non-Hermitian linear systems with multiple right-hand sides , 1997 .

[7]  M. Sadkane.  Block-Arnoldi and Davidson methods for unsymmetric large eigenvalue problems , 1993 .

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

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

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

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

[12]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[13]  Qiang Ye,et al.  ABLE: An Adaptive Block Lanczos Method for Non-Hermitian Eigenvalue Problems , 1999, SIAM J. Matrix Anal. Appl..

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

[15]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[16]  Y. Saad,et al.  Numerical solution of large Lyapunov equations , 1989 .

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

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

[19]  Daniel Boley.  Krylov space methods on state-space control models , 1994 .

[20]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

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

[22]  D. Hu,et al.  Krylov-Subspace Methods for the Sylvester Equation , 2001 .

[23]  G. Stewart.  Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems , 1973 .

[24]  C. Loan,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix , 1978 .