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Approximate Solutions to Large Stein Matrix Equations

Authors: Khalide Jbilou

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: journal, paper, latex, template, IEEEtran

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1061126

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[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.