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

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.