A minimal residual algorithm for the inconsistent matrix equation AXB=C over symmetric matrices

Abstract An iterative method with short recurrences is presented by Peng [Z.-Y. Peng, An iteration method for the least squares symmetric solution of the linear matrix equation AXB = C, Appl. Math. Comput. 170 (2005) 711–723] for solving the nearness problem associated with the inconsistent matrix equation AXB = C for symmetric matrices. The solution of this nearness problem can be computed with little work and low storage requirements per iteration. However, this algorithm may be slow in case of the irregular convergence behavior in the residual norm of AXB = C. In order to remedy this problem, a modification based on the idea of the classical CG method is presented in this paper and an error bound is given. Finally, numerical experiments are reported.

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

[2]  Xi-Yan Hu,et al.  An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C , 2005, Appl. Math. Comput..

[3]  O. Axelsson.  Iterative solution methods , 1995 .

[4]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[5]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[6]  Y. G. Tsuei,et al.  Estimation of system matrices by dynamic condensation and application to structural modification , 1995 .

[7]  Zhen-yun Peng.  An iterative method for the least squares symmetric solution of the linear matrix equation AXB = C , 2005, Appl. Math. Comput..

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

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

[10]  Anping,et al.  OPTIMAL APPROXIMATE SOLUTION OF THE MATRIX EQUATION AXB = C OVER SYMMETRIC MATRICES , 2007 .

[11]  P.B. Kis.  Modeling of dynamical systems and applications , 2005, 2005 IEEE International Conference on Intelligent Engineering Systems, 2005. INES '05..

[12]  N. Higham.  MATRIX NEARNESS PROBLEMS AND APPLICATIONS , 1989 .

[13]  John E. Mottershead,et al.  Finite Element Model Updating in Structural Dynamics , 1995 .

[14]  N. Higham.  Applications of matrix theory , edited by M. J. C. Gover and S. Barnett. Pp 324. £40. 1989. ISBN 0-19-853625-9 (Oxford University Press) , 1990, The Mathematical Gazette.

[15]  Kevin Hayden,et al.  Modeling of dynamical systems , 2007 .