FINITE ITERATIVE ALGORITHM FOR THE COMPLEX GENERALIZED SYLVESTER TENSOR EQUATIONS*

A finite iterative algorithm is proposed to solve a class of complex generalized Sylvester tensor equations. The properties of this proposed algorithm are discussed based on a real inner product of two complex tensors and the finite convergence of this algorithm is obtained. Two numerical examples are offered to illustrate the effectiveness of the proposed algorithm.


Introduction
A tensor is a multidimensional array of data whose elements are referred by using multiple indices. We use to denote a complex tensor A, where N is called the order of tensor A and (I 1 , I 2 , . . . , I N ) is the dimension of A. As a special case, the vector is a 1-order tensor and the matrix is a 2-order tensor. Let A = (a i1i2···i N ) ∈ C I1×I2×···×I N be a conjugate of tensor A, where a i1i2···i N is the complex conjugate of a i1i2···i N .
In this paper, we consider the iterative algorithm to solve the following complex generalized Sylvester tensor equation X × 1 A 1 + · · · + X × N A N + X × 1 B 1 + · · · + X × N B N = D, (1.1) where the matrices A n , B n ∈ C In×In (n = 1, 2, . . . , N ) and the right-side tensor D ∈ C I1×I2×···×I N are known, and X ∈ C I1×I2×···×I N is the unknown tensor to be solved. For the tensor X ∈ C I1×I2×···×I N and the matrix U ∈ C In×In , X × n U ∈ C I1×I2×···×I N is defined by When the complex generalized tensor equation (1.1) over the real field R and B n = 0 for all n = 1, 2, . . . , N , the tensor equation (1.1) reduces to the real Sylvester tensor equation (1.2) By using the Kronecker product [1,13], the tensor equation (1.2) can be reformulated as follows: with A = I (I N ) ⊗ · · · ⊗ I (I2) ⊗ A 1 + · · · + A N ⊗ I (I N −1 ) ⊗ · · · ⊗ I (I1) , (1.4) and x = vec(X ), b = vec(D). Here, I (n) stands for the identity matrix of order n and the operator vec(·) stacks the columns of a tensor to form a vector. In the case that X is a 3-order tensor, the tensor equation (1.2) reduces to which comes from the finite element [7] or spectral method [14,[16][17][18] discretization of a linear partial differential equation in high dimension. When X is a simple 2-order tensor, the complex generalized tensor equation (1.1) reduces to the extended Sylvester-conjugate matrix equation and the Sylvester tensor equation (1.2) reduces to the Sylvester matrix equation which arises frequently from the areas of systems and control theory [6] and has received much attention, e.g., see [5,[8][9][10]. Nowadays, tensor equations have attracted much attention [1][2][3][4]7,11,13,15]. For example, Kressner and Tobler in [13] proposed a so-called tensor Krylov subspace method for solving (1.3) with the coefficient matrix A as (1.4) and the right-hand side b = b 1 ⊗b 2 ⊗· · ·⊗b N , which is based on a tensorized Krylov subspace associated with A and b, i.e., the Kronecker product of the usual Krylov subspaces. The tensorized Krylov subspace method transforms the system (1.3) to a system of smaller size. In [1], Ballani and Grasedyck presented an iterative scheme similarly to Krylov subspace method to solve the linear system (1.3) with which is based on the projection of the residual to a low dimensional subspace and all calculations are performed in hierarchical Tucker format. In particular, Beik, Movahed and Ahmadi-Asl in [2] proposed the conjugate gradient algorithm of tensor form (CG-BTF) to solving the Sylvester tensor equation ( 1, 2, . . . , p; i 2 = 1, 2, . . . , q), F ∈ C m×n are the given known matrices and X ∈ C r×s is the matrix to be determined, Wu, Lv and Hou [19] presented an iterative algorithm for solving (1.8). After that, Zhang in [21] offered the finite iterative algorithm for solving the complex generalized coupled Sylvester matrix equations, that is . . . , p; j = 1, 2, . . . , q) are the given known matrices and X j ∈ C rj ×sj (j = 1, 2, . . . , q) are the unknown matrices. In this paper, the finite iterative algorithm for solving the complex generalized Sylvester tensor equation (1.1) will be discussed. The remainder of this paper is organized as follows. In Section 2, we offers some symbols and preliminaries. In Section 3, we present a finite iterative algorithm for solving the complex generalized Sylvester tensor equation (1.1) and constructs several results which be used in the convergence proof. Numerical examples are offered in Section 4 to illustrate the effectiveness of the proposed algorithm. Finally, some concluding remarks are given in Section 5.

Preliminaries
For a matrix A = (a ij ) ∈ C I×I , the symbols A T and A H denote the transpose and conjugate transpose of matrix A, that is Now, we recall the real inner product, which is given for complex spaces over the real field R. This inner product will play a very important role in this paper.

Definition 2.1 ( [20]
). A real inner product space is a vector space V over the real field R together with an inner product, i.e., with a mapping ⟨·, ·⟩ r : V × V → R satisfying the following three axioms for all vectors x, y, z ∈ V and all scalars η ∈ R.
For the tensors X , Y ∈ C I1×I2×···×I N , the inner product is defined as and a real inner product can be defined as where Re(λ) denotes the real part of the complex number λ. In fact, for the tensors The induced-norm of a tensor X is defined by the following formula For the linear operator L from C I1×I2×···×I N to C I1×I2×···×I N , the conjugate linear operator L * from C I1×I2×···×I N to C I1×I2×···×I N satisfies according to the conjugate operator, we have Proof. By the computation, we have For simplicity, in the sequel, we use the following linear operator: Then, the complex generalized Sylvester tensor equation (1.1) can be reformulated by According to Lemma 2.1, it is not difficult to verify that L * is specified as follows: With these results, we will present a finite iterative algorithm to solve the complex generalized Sylvester tensor equation (1.1) in the following section.

Algorithm and convergence
In this section, we will propose a finite iterative algorithm for solving the complex generalized Sylvester tensor equation (1.1) and establish the convergence result for the proposed iterative algorithm. Step 1. Let the matrices A n , B n ∈ C In×In for n = 1, 2, . . . , N and the right-hand side D. Choose any initial tensor X 0 ∈ C I1×I2×···×I N , compute Step 2. If ∥R k ∥ = 0 or ∥Q k ∥ = 0, then stop; else compute Step 3. Set k := k + 1, go to Step 2. Now, we give some results for Algorithm 3.1.
Theorem 3.1. According to Algorithm 3.1, the tensor sequences R k and Q k for k = 0, 1, 2, . . . satisfy Proof. Using the mathematical induction. Firstly, we prove that In fact, we have Now, we assume that the results of Theorem 3.1 are correct for k, l ≤ s(s > 1). For l = s + 1 and k = s, we have Finally, we prove that the results of Theorem 3.1 are correct for l = s + 1 and k ≤ s − 1. In fact, we have According to the mathematical induction, the proof is completed.

Theorem 3.2.
If the tensor X * is a solution of the complex generalized Sylvester tensor equation (1.1), for any initial iterative tensor X 0 ∈ C I1×I2×···×I N , the tensor sequences R k and Q k generated by Algorithm 3.1 satisfy Proof. Using the mathematical induction. Firstly, we verify that In fact, we have Now, we assume that the result of Theorem 3.2 is correct for k = s(s ≥ 1). For k = s + 1, we have According to the mathematical induction, the proof is completed. With Theorems 3.1 and 3.2, we can obtain the following conclusion.

Theorem 3.3. If the complex generalized Sylvester tensor equation (1.1) is consistent, then a solution can be obtained within finite iteration steps by using Algorithm 3.1 for any initial tensor
At the end of this section, we present the NCG-BTF algorithm [2] for the Sylvester tensor equation (1.2) with A n ∈ R In×In (n = 1, 2, . . . , N ) being real nonsymmetric positive definite. Let for n = 1, 2, . . . , N . Substituting A n = H n − S n into (1.2), we have We compute X k+1 as the solution of the following tensor equation: where X k is the k-th approximate solution to the exact solution X * of (1.2). Notice that for the tensors X , Y ∈ R I1×I2×···×I N , the inner product is reduced to Then, the NCG-BTF algorithm for solving (1.2) is as follows.

Numerical examples
In this section, we use some test problems to examine the numerical effectiveness of Algorithm 3.1 for solving the complex generalized Sylvester tensor equation (1.1).
In actual computations, all runs are started from the initial tensor X 0 = O, are terminated if the current iterations satisfy (i) ∥R k ∥ ≤ 10 −6 or ∥Q k ∥ ≤ 10 −6 for Algorithm 3.1, (ii) ∥R k ∥ ≤ 10 −6 for Algorithm 3.2, or if the number of the prescribed iterative steps k max = 4000 is exceeded, and are performed on a personal computer with 3.60 GHz central processing unit (Intel(R) Core(TM) i7-7700), 32.0 GB memory and Windows 10 operating system.
Here, 'IT' and 'CPU' denote the number of iteration steps and elapsed CPU time in seconds, respectively. In addition, ERR := ∥R k ∥. We comment here that the Tensor Toolbox [12] is utilized for solving the succeeding discussed problems Example 4.1. Consider the following convection-diffusion equation By using a standard finite difference discretization on a uniform grid for the diffusion term and a second-order convergent scheme (Fromm's scheme) for the convection term with the mesh-size h = 1/(p + 1), the discrete system matrix of the form (1.2) is obtained such that And the right-hand side D is constructed so that X * = ones(p, p, p) is the exact solution of the real Sylvester tensor equation (1.2).
For Example 4.1, we compare Algorithms 3.1 and 3.2. The inner iteration of Algorithm 3.2 is terminated if the value σ = 10 −3 or the number of the prescribed inner iterative steps ℓ max = 100. The notation '-' denotes the case that the algorithm is stopped after k max iterations without computing a suitable approximate solution.
The obtained results are presented in Table 1 with c 1 = c 2 = c 3 = 1 and Table  2 with c 1 = 2, c 2 = 4 and c 3 = 8, respectively. From both tables, we can see that the NCG BTF algorithm is stopped after k max iterations while the proposed finite iterative algorithm performs well for the viscous parameter ν = 0.01, 0.1. Notice that when ν = 1 and c 1 = c 2 = c 3 = 1, the NCG BTF algorithm surpasses the finite iterative algorithm in CPU time. However, the finite iterative algorithm outperforms the NCG BTF algorithm in terms of the required CPU time when ν = 1, c 1 = 2, c 2 = 4 and c 3 = 8.
For Example 4.2, we get the approximate solution as X * after 312 iterative steps, which the residual and error are ERR = 6.1095e-05 and ∥X 312 − X * ∥ = 2.2969e-06.
For more details, the convergence history of the finite iterative algorithm is plotted in Figure 1.

Conclusions
We have proposed a finite iterative algorithm to solve the complex generalized Sylvester tensor equation (1.1), which is generalized of Sylvester tensor equations (1.2) considered in [2]. It is proved that the algorithm is convergent within the finite iterative steps in the absence of the round-off error, which is based on the introducing real inner product. The applications of the proposed algorithm have been compared with the NCG-BTF algorithm when solves the Sylvester tensor equation (1.2). In addition, another example is offered to illustrate the effectiveness of the proposed algorithm for the complex generalized Sylvester tensor equation (1.1).