Global least squares methods based on tensor form to solve a class of generalized Sylvester tensor equations

Abstract

This paper is concerned with some of well-known iterative methods in their tensor forms to solve a class of tensor equations via the Einstein product and the associated with least squares problem. Especially, the tensor forms of the LSQR and LSMR methods are presented. The proposed methods use tensor computations with no matricizations involved. We prove that the norm of residual is monotonically decreasing for the tensor form of the LSQR method. The norm of residual of normal equation is also monotonically decreasing for the tensor form of the LSMR method. We also show that the minimum-norm solution (or the minimum-norm least squares solution) of the tensor equation can be obtained by the proposed methods. Numerical examples are provided to illustrate the efficiency of the proposed methods and testify the conclusions suggested in this paper.