An iterative algorithm to solve the generalized Sylvester tensor equations

This paper is concerned with the conjugate gradient least squares algorithm to solve a class of tensor equations via the Einstein product. The proposed algorithm uses tensor computations with no matricizations involved. We prove that the solution (or the least squares solution) of the tensor equation can be obtained within a finite number of iterative steps in the absence of round-off errors. By selecting the appropriate initial tensor, the least Frobenius norm solution (or the least Frobenius norm least squares solution) of the tensor equation can be obtained. Numerical examples are provided to illustrate the efficiency of the proposed algorithm and testify the conclusions suggested in this paper.