On the Execution and Convergence of GMRES − BTF Method for Solving Sylvester Tensor Equations Fatemeh Panjeh Ali Beik , Academic member of Department of Mathematics, f.beik@vru.ac.ir Department of Mathematics, Vali-e-Asr University of Rafsanjan, P. O. Box 518, Rafsanjan, Iran. Salman Ahmadi-Asl * , ** Ph.D. student, p92357001@post.vru.ac.ir Department of Mathematics, Vali-e-Asr University of Rafsanjan, P. O. Box 518, Rafsanjan, Iran. Abstract: Recently, Chen and Lu have handled the well-known generalized minimal residual based on tensor format (GMRES − BTF) for solving the Sylvester tensor equation. Nevertheless, the construction and convergence of the presented algorithm have not been discussed theoretically. This fact inspirits us to theoretically analyze the construction of the GMRES − BTF method. To this end, we introduce a new product between two tensors and elaborate some of its properties. The presented product can be exploited to illustrate that how the GMRES − BTF algorithm is implemented in the tensor format and to establish the convergence properties of the algorithm. Keywords: Sylvester tensor equation; GMRES; Iterative method; Convergence. 1 INTRODUCTION It is known that a tensor is a multidimensional ar- ray. In [4], the authors have elaborated an overview of higher-order tensors and their decomposition. The order of a tensor is the number of dimensions which is called by modes or ways. During this pa- per, vectors (tensors of order one) are represented by lowercase letter, matrices (tensors of order two) are signified by capital letters. Higher-order ten- sors (order three or higher) are indicated by Euler script letters, e.g., X. The current paper deals with the solution of the next Sylvester tensor equation X × 1 A (1) + X × 2 A (2) + ... + X × N A (N) = D, (1) where the matrices A (n) ∈ R In×In (n =1, 2,...,N ) and tensor D ∈ R I 1 ×I 2 ×...×I N are known and X is the unknown tensor. The definition and properties of the n-mode product × n are expounded before ending this section. It is not difficult to verify that (1) is equivalent to the following linear system of equations Ax = b, (2) with x = vec(X), b = vec(D) and A = N ∑ j=1 I (I N ) ⊗ ... ⊗ I (I j+1 ) ⊗ A (j) ⊗ I (I j-1 ) ⊗ ... ⊗ I (I 1 ) where ⊗ denotes the Kronecker product, I (n) stands for the identity matrix of order n and the operator ”vec” stacks the column of a matrix (or a tensor) to form a vector. It is well-known that the linear system (2) has a unique solution if and only if the coefficient matrix A is nonsingular. In this paper the matrix A is assumed to be nonsingu- lar. In [2], a gradient based approach [3] has been * Corresponding Author 1