Approximate Solutions to Large Stein Matrix Equations
In the present paper, we propose numerical methods for solving the Stein equation AXC - X - D = 0 where the matrix A is large and sparse. Such problems appear in discrete-time control problems, filtering and image restoration. We consider the case where the matrix D is of full rank and the case where D is factored as a product of two matrices. The proposed methods are Krylov subspace methods based on the block Arnoldi algorithm. We give theoretical results and we report some numerical experiments.
[1] H. Kopka and P. W. Daly, A Guide to LATEX, 3rd ed. Harlow, England:
Addison-Wesley, 1999.
[2] A. Y. BARRAUD, A numerical algorithm to solve ATXA − X = Q,
IEEE Trans. Autom. Contr., AC-22(1977), pp. 883-885.
[3] R. H. BARTELS AND G. W. STEWART, Algorithm 432: Solution of the
matrix equation AX+XB=C, Circ. Syst. and Signal Proc., 13(1994), pp.
820-826.
[4] A. BOUHAMIDI AND K. JBILOU, Sylvester Tikhonov-regularization methods
in image restoration, J. Comput. Appl. Math., to appear.
[5] B.N. DATTA, Numerical Methods for Linear Control Systems, Elsevier
Academic press, 2004.
[6] B.N. DATTA, Krylov-subspace methods for large scale matrix problems
in control, Generation of Computer Systems, 19(2003), pp. 125-126.
[7] A. EL GUENNOUNI, K. JBILOU AND A.J. RIQUET , Block Krylov
subspace methods for solving large Sylvester equations, Numer. Alg.,
29(2002), pp. 75-96.
[8] K. GLOVER, D.J.N. LIMEBEER, J.C. DOYLE, E.M. KASENALLY AND
M.G. SAFONOV, A characterisation of all solutions to the four block
general distance problem, SIAM J. Control Optim., 29(1991), pp. 283-
324.
[9] G. H. GOLUB, S.NASH AND C. VAN LOAN, A Hessenberg-Schur method
for the problem AX+XB=C, IEEC Trans. Autom. Contr., AC-24(1979),
pp. 909-913.
[10] I. M. JAIMOUKHA AND E. M. KASENALLY, Krylov subspace methods
for solving large Lyapunov equations, SIAM J. Numer. Anal., 31(1994),
pp. 227-251.
[11] P. LANCASTER AND L. RODMAN, The Algebraic Riccati Equations,
Clarendon Press, Oxford 1995.
[12] A. RUHE, Rational Krylov sequence methods for eigenvalue computations,
Linear Algebra Appl., 58(1984), pp. 391-405.
[13] Y. SAAD AND M. H. SCHULTZ, GMRES: A generalized minimal
residual algorithm for solving nonsymmetric linear systems, SIAMJ. Sci.
Statis. Comput., 7(1986), pp. 856-869.
[14] Y. SAAD, Iterative Methods for Sparse Linear Systems, PWS Press, New
York, 1995.
[15] M. SADKANE, Block Arnoldi and Davidson methods for unsymmetric
large eigenvalue problems, Numer. Math., 64(1993), pp. 687-706.
[16] R. SMITH, Matrix equation XA+BX=C, SIAM J. Appl. Math.,
16(1)(1968), pp. 198-201.
[17] P. VAN DOOREN, Gramian based model reduction of large-scale dynamical
systems , in Numerical Analysis, Chapman and Hall, pp. 231-247,
CRC Press London, 2000.
[1] H. Kopka and P. W. Daly, A Guide to LATEX, 3rd ed. Harlow, England:
Addison-Wesley, 1999.
[2] A. Y. BARRAUD, A numerical algorithm to solve ATXA − X = Q,
IEEE Trans. Autom. Contr., AC-22(1977), pp. 883-885.
[3] R. H. BARTELS AND G. W. STEWART, Algorithm 432: Solution of the
matrix equation AX+XB=C, Circ. Syst. and Signal Proc., 13(1994), pp.
820-826.
[4] A. BOUHAMIDI AND K. JBILOU, Sylvester Tikhonov-regularization methods
in image restoration, J. Comput. Appl. Math., to appear.
[5] B.N. DATTA, Numerical Methods for Linear Control Systems, Elsevier
Academic press, 2004.
[6] B.N. DATTA, Krylov-subspace methods for large scale matrix problems
in control, Generation of Computer Systems, 19(2003), pp. 125-126.
[7] A. EL GUENNOUNI, K. JBILOU AND A.J. RIQUET , Block Krylov
subspace methods for solving large Sylvester equations, Numer. Alg.,
29(2002), pp. 75-96.
[8] K. GLOVER, D.J.N. LIMEBEER, J.C. DOYLE, E.M. KASENALLY AND
M.G. SAFONOV, A characterisation of all solutions to the four block
general distance problem, SIAM J. Control Optim., 29(1991), pp. 283-
324.
[9] G. H. GOLUB, S.NASH AND C. VAN LOAN, A Hessenberg-Schur method
for the problem AX+XB=C, IEEC Trans. Autom. Contr., AC-24(1979),
pp. 909-913.
[10] I. M. JAIMOUKHA AND E. M. KASENALLY, Krylov subspace methods
for solving large Lyapunov equations, SIAM J. Numer. Anal., 31(1994),
pp. 227-251.
[11] P. LANCASTER AND L. RODMAN, The Algebraic Riccati Equations,
Clarendon Press, Oxford 1995.
[12] A. RUHE, Rational Krylov sequence methods for eigenvalue computations,
Linear Algebra Appl., 58(1984), pp. 391-405.
[13] Y. SAAD AND M. H. SCHULTZ, GMRES: A generalized minimal
residual algorithm for solving nonsymmetric linear systems, SIAMJ. Sci.
Statis. Comput., 7(1986), pp. 856-869.
[14] Y. SAAD, Iterative Methods for Sparse Linear Systems, PWS Press, New
York, 1995.
[15] M. SADKANE, Block Arnoldi and Davidson methods for unsymmetric
large eigenvalue problems, Numer. Math., 64(1993), pp. 687-706.
[16] R. SMITH, Matrix equation XA+BX=C, SIAM J. Appl. Math.,
16(1)(1968), pp. 198-201.
[17] P. VAN DOOREN, Gramian based model reduction of large-scale dynamical
systems , in Numerical Analysis, Chapman and Hall, pp. 231-247,
CRC Press London, 2000.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:54572", author = "Khalide Jbilou", title = "Approximate Solutions to Large Stein Matrix Equations", abstract = "In the present paper, we propose numerical methods for solving the Stein equation AXC - X - D = 0 where the matrix A is large and sparse. Such problems appear in discrete-time control problems, filtering and image restoration. We consider the case where the matrix D is of full rank and the case where D is factored as a product of two matrices. The proposed methods are Krylov subspace methods based on the block Arnoldi algorithm. We give theoretical results and we report some numerical experiments.
", keywords = "IEEEtran, journal, LATEX, paper, template.", volume = "6", number = "10", pages = "1406-5", }
