A note on the iterative solutions of general coupled matrix equation

Abstract Recently, Ding and Chen [F. Ding, T. Chen, On iterative solutions of general coupled matrix equations, SIAM J. Control Optim. 44 (2006) 2269–2284] developed a gradient-based iterative method for solving a class of coupled Sylvester matrix equations. The basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified, so that the iterative solutions are obtained by applying hierarchical identification principle. In this note, by considering the coupled Sylvester matrix equation as a linear operator equation we give a natural way to derive this algorithm. We also propose some faster algorithms and present some numerical results.

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

[2]  Sang-Yeun Shim,et al.  LEAST SQUARES SOLUTION OF MATRIX EQUATION , 2003 .

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

[4]  Zhong-Zhi Bai,et al.  The Constrained Solutions of Two Matrix Equations , 2002 .

[5]  Feng Ding,et al.  On Iterative Solutions of General Coupled Matrix Equations , 2006, SIAM J. Control. Optim..

[6]  Zhong-Zhi Bai,et al.  Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations , 2006, Numer. Linear Algebra Appl..

[7]  Z. Bai.  ON HERMITIAN AND SKEW-HERMITIAN SPLITTING ITERATION METHODS FOR CONTINUOUS SYLVESTER EQUATIONS * , 2010 .

[8]  Daniel Boley,et al.  Numerical Methods for Linear Control Systems , 1994 .

[9]  Marlis Hochbruck,et al.  Preconditioned Krylov Subspace Methods for Lyapunov Matrix Equations , 1995, SIAM J. Matrix Anal. Appl..

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

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

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

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

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

[15]  W. Niethammer,et al.  SOR for AX−XB=C , 1991 .

[16]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[17]  Feng Ding,et al.  Gradient Based Iterative Algorithms for Solving a Class of Matrix Equations , 2005, IEEE Trans. Autom. Control..

[18]  Bruce A. Francis,et al.  Optimal Sampled-Data Control Systems , 1996, Communications and Control Engineering Series.

[19]  Lothar Reichel,et al.  Application of ADI Iterative Methods to the Restoration of Noisy Images , 1996, SIAM J. Matrix Anal. Appl..

[20]  Gene H. Golub,et al.  Matrix computations , 1983 .

[21]  Khalide Jbilou,et al.  ADI preconditioned Krylov methods for large Lyapunov matrix equations , 2010 .

[22]  Yuan Lei,et al.  Best Approximate Solution of Matrix Equation AXB+CYD=E , 2005, SIAM J. Matrix Anal. Appl..

[23]  AnpingLiao,et al.  LEAST—SQUARES SOLUTION OF AXB=D OVER SYMMETRIC POSITIVE SEMIDEFINITE MATRICES X , 2003 .

[24]  Hiroaki Mukaidani,et al.  New iterative algorithm for algebraic Riccati equation related to H ∞ control problem of singularly perturbed systems , 2001, IEEE Trans. Autom. Control..

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

[26]  Feng Ding,et al.  Iterative least-squares solutions of coupled Sylvester matrix equations , 2005, Syst. Control. Lett..

[27]  Peter Benner,et al.  On the ADI method for Sylvester equations , 2009, J. Comput. Appl. Math..