Extending the CGLS algorithm for least squares solutions of the generalized Sylvester-transpose matrix equations

This paper deals with the solution to the least squares problemminX‖∑i=1sAiXBi+∑j=1tCjXTDj−E‖,corresponding to the generalized Sylvester-transpose matrix equation. The conjugate gradient least squares (CGLS) method is extended to obtain a matrix algorithm for solving this problem. We show that the matrix algorithm can solve this problem within a finite number of iterations in the absence of roundoff errors. Also the descent property of the norm of residuals is obtained. Finally numerical results demonstrate the accuracy and robustness of the algorithm.