Extended conjugate gradient squared and conjugate residual squared methods for solving the generalized coupled Sylvester tensor equations

Abstract

In this paper, two attractive iterative methods – conjugate gradient squared (CGS) and conjugate residual squared (CRS) – are extended to solve the generalized coupled Sylvester tensor equations $\sum _{j=1}^n{\mathcal{X}}_j{\times }_1{A}_{\mathrm{ij}1}{\times }_2{A}_{\mathrm{ij}2}...{\times }_d{A}_{\mathrm{ijd}}={\mathcal{C}}_i,i=1,2,...,n$ . The proposed methods use tensor computations with no maricizations involved. Also, some properties of the new methods are presented. Finally, several numerical examples are given to compare the efficiency and performance of the proposed methods with some existing algorithms.