Solving Two Coupled Fuzzy Sylvester Matrix Equations Using Iterative Least-squares Solutions

ABSTRACT In this paper, five iterative methods for solving two coupled fuzzy Sylvester matrix equations are considered. The two coupled fuzzy Sylvester matrix equations are expressed by using the generalized inverse of the coefficient matrix, then iterative solutions are constructed by applying the hierarchical identification principle and by using the block-matrix inner product (the star product for short). A proposed modification to this algorithm to solve the first coupled fuzzy Sylvester matrix equations is suggested. This proposed modification is compared with the first algorithm where our modification exhibits fast convergence behavior. Also, we suggested two least-squares iterative algorithm by applying a hierarchical identification principle to solve the two coupled fuzzy Sylvester matrix equations. The proposed methods are illustrated by numerical examples.


Introduction
Many authors attempt to solve coupled Sylvester matrix equations by various methods. Ding et al. [1] obtained the approximate solutions of the matrix equation A X B = F and the generalized Sylvester matrix equation A X B + C X D = F, by extending Jacobi and Gauss-Seidel iteration methods for A x = b. Ding and Chen [2] suggested a least-squares iterative algorithm to solve the generalized coupled Sylvester matrix equation In [3], a large family of iterative methods to solve coupled Sylvester matrix equations by applying a hierarchical identification principle is presented. Iterative algorithms for obtaining the unique least-squares solution were proposed in [2,3] by introducing the block-matrix inner product. Efficient numerical algorithms are presented with the gradient-based iterative algorithms [3,4] and least square-based iterative algorithms [3] for solving coupled matrix equations. Hajarian [5] suggested a conjugate direction (CD) algorithm to find the generalized reflexive solution X and the generalized anti-reflexive solution Y of the coupled Sylvester matrix equations Zhang [6] constructed a gradient-based iterative algorithm to solve the real coupled matrix equations (2) by using the hierarchical identification principle. Bayoumi et al. [7] suggested a modified gradient based iterative algorithm for solving extended Sylvester-conjugate matrix equations AXB + CXD = F. Friedman et al. [8] proposed a general model for solving an n × n fuzzy linear system with a crisp coefficient and an arbitrary vector of fuzzy numbers on the right-hand side column. In [9], fuzzy numbers with a new parametric form are presented. And a new fuzzy arithmetic is defined and applied to fuzzy linear equations and fuzzy calculus. In [10], the common solution pair of fuzzy matrix equations is studied and the Kronecker product and Vec-operator for transforming the system of fuzzy linear matrix equation to a fuzzy linear system are employed. Bayoumi [11] proposed finite iterative Hamiltonian solutions of the generalized coupled Sylvester-conjugate matrix equations. Bayoumi and Ramadan [12] introduced finite iterative Hermitian R-conjugate solutions of the generalized coupled Sylvester-conjugate matrix equations. Behera and Chakraverty [13] proposed a new and simple method to solve fuzzy real system of linear equations with Crisp Coefficients. Wang et al. [14] investigated the least-squares solution with the least norm to a system of tensor equations over the quaternion algebra.
This paper is organized as follows: first, in Section 2, we introduce some notations, definitions, lemmas and theorems that will be needed to develop this work. In Section 3, we suggest five iterative algorithms to obtain the solutions of two coupled fuzzy Sylvester matrix equations. In first algorithm, we investigate the coupled fuzzy Sylvester matrix equations given in (1) by using the generalized inverse of the coefficient matrix, then iterative solutions are constructed by applying the hierarchical identification principle and by using the block-matrix inner product, and we propose a modification to this algorithm in the second algorithm for the same matrix equations. In third algorithm, we introduce least-squares iterative algorithm by applying a hierarchical identification principle to solve coupled fuzzy Sylvester matrix equations given in (1). In fourth algorithm, we investigate the coupled fuzzy Sylvester matrix equations given in (2) by using the generalized inverse of the coefficient matrix, then iterative solutions are constructed by applying the hierarchical identification principle and by using the block-matrix inner product. In fifth algorithm, we introduce least-squares iterative algorithm by applying a hierarchical identification principle to solve coupled fuzzy Sylvester matrix equations given in (2). And we give the convergence properties of these iterative algorithms. In Section4, numerical examples are introduced to illustrate the effectiveness of the proposed algorithms.

Preliminaries
The following notations, definitions, lemmas and theorems will be used to develop the proposed work. We use A T to denote the transpose of A. The set of all m × n real matrices is denoted by R m×n . For A ∈ R m×n , vec (A) is defined as vec (A) = [a T 1 a T 2 · · · · · · a T n ] T where a i is the ith column of the matrix A. The Kronecker product of two matrices A = (a ij ) m×n and B is denoted by A ⊗ B. We have the following well-known property vec (M X N) = (N T ⊗ M) vec (X) for matrices M, X, N. Definition 2.1: Block-matrix inner product [2] The block-matrix inner product is called the star product for short, denoted by ( * ). Let Then the block-matrix star product is defined as The following basic concepts of fuzzy number arithmetic and fuzzy linear system of equations will be used to develop the proposed work.

Definition 2.2: Fuzzy number [8]
A fuzzy number in parametric form is an ordered pair of functions (u(r), u(r)), 0 ≤ r ≤ 1, which satisfies the following requirements: The triangular fuzzy numbers are very popular and denoted by u = (c, α, β) and defined by where α > 0 and β > 0. The parametric form of the number is u(r) = rα + c − α, The addition and scalar multiplication of fuzzy numbers are defined by the extension principle and can be equivalently represented as follows, see [8,9].

Definition 2.3:
Consider the p × q linear system of equations where the coefficient matrix A = (a ij ) ∈R p×q is given crisp matrix and w = (w 1 , w 2 , . . . , w p ) T is given vector of fuzzy numbers and v = (v 1 , v 2 , . . . , v q ) T is vector of fuzzy numbers to be determined. This system is called an FSLE.

Definition 2.4: A fuzzy number vector
is called a solution of the fuzzy linear system of equations (3) if In general, an arbitrary equation for either w i or w i is a linear combination of v j 's and v j 's, respectively. Therefore, in order to solve Equation (3) one must solve a 2p × 2q crisp linear system of equations (5) as follows: where the element of S = (s ij ), 1 ≤ i, j ≤ 2q, as follows: Theorem 2.1: [10]: Let matrix S be in the form (6), then the matrix and (S 1 − S 2 ), respectively. In particular, the Moore-Penrose inverse of the matrix S is Theorem 2.2: [10]: For the consistent system (5) and any {1, w is a solution to the system (5). Lemma 2.1: [15] For matrix equation Ax = b, if A is a full column-rank matrix, then the following least squares based iterative algorithm leads to lim k→∞ x(k) = x Lemma 2.2: [15] For matrix equation AXB = F, if A is a full column-rank matrix and B is a full row-rank matrix, then the iterative solution X(k) given by the following least squares based iterative algorithm converges to the exact solution X for any initial values X(0):

Lemma 2.3: [2]
The coupled fuzzy Sylvester matrix equations given in (1), where A, D ∈R m×m and B, E ∈R n×n are given crisp matrices and C, F ∈R m×n are given fuzzy matrices while X, Y ∈R m×n are fuzzy matrices to be determined. Equation (1) has a unique solution if and only if the matrix is non-singular; in this case, the unique solution is given by and the corresponding homogeneous matrix equation AX + YB = 0 , DX + YE = 0 has a unique solution X = Y = 0.

Lemma 2.4:
The coupled Sylvester matrix equations given in (2), where A, C, E, H ∈R m×m and B, D, G, N ∈R n×l are given crisp matrices and F 1 , F 2 ∈R m×l are given fuzzy matrices while X, Y ∈R m×n are fuzzy matrices to be determined. Equation (2) has a unique solution if and only if the matrix is non-singular; in this case, the unique solution is given by and the corresponding homogeneous matrix equation AXB + CYD = 0, EXG + HYN = 0 has a unique solution X = Y = 0.

The Main Results
In this section, we consider five iterative algorithms to solve two coupled fuzzy Sylvester matrix equations. Algorithm I and algorithm IV adopt the line of the one in [2].

Iterative Algorithm for Solving the Coupled Fuzzy Sylvester matrix equations (1)
In this section, we present an iterative least-squares algorithm for solving coupled fuzzy Sylvester matrix equations given in (1), where A, D ∈R m×m and B, E ∈R n×n are given crisp matrices and C, F ∈ R m×n are given fuzzy matrices while X, Y ∈ R m×n are fuzzy matrices to be determined. The basic idea is to regard Equation (1) as two matrices Hence, Equation (1) can be decomposed into two matrix equations of the form: Here Then, we can define the following iterative formulas where μ is the convergence factor. Substituting from Equations (10) and (11) into Equations (14) and (15) gives The right-hand sides of these equations include the unknown fuzzy matrices X and Y, so it is impossible to realize the algorithm in Equations (16) and (17). By applying the hierarchical identification principle [4], the unknown fuzzy matrices X and Y in these equations is replaced with its estimate X(k) and Y(k). Thus one has In this case, the iterative least-squares solutions of coupled fuzzy Sylvester matrix equations can be written as We now outline our suggested algorithm.

Algorithm I
Step 1 Input crisp matrices A, D ∈R m×m and B, E ∈R n×n and input fuzzy matrices C, F ∈R m×n , and number ε.
Proof: First, we define the estimation error matrices as Using algorithm I and the above error matrices, we can obtain Now, by taking the norm of (25) and (26) and using the following formula, we have Gives Similarly, Define the nonnegative definite function η(k) by From (28) and (29), this function can be computed as If the convergence factor μ is chosen to satisfy Since the matrix equation (1) has a unique fuzzy solution it follows that as k → ∞

A Modified Iterative Algorithm to Solve the Coupled Fuzzy Sylvester Matrix Equations (1)
In this subsection, we propose a modification to algorithm I to solve coupled fuzzy Sylvester matrix equations given in (1), where A, D ∈R m×m and B, E ∈R n×n are given crisp matrices and C, F ∈R m×n are given fuzzy matrices while X, Y ∈R m×n are fuzzy matrices to be determined. The proposed algorithm is as follows: Algorithm 1: Step 1 Input crisp matrices A, D ∈R m×m and B, E ∈R n×n and input fuzzy matrices C, F ∈R m×n , and number ε.
Step 7 End Note that in the step of computing Y(k), the last approximate solution X(k) has been computed. Hence, we can use the information of X(k) to update the Y(k). Similarly, in the step of computing Y(k), the last approximate solution X(k) has been computed. Hence, we can use the information of X(k) to update the Y(k).

Least Squares Based Iterative Solutions of Coupled Fuzzy Sylvester Matrix Equations (1)
In this section, we are studying the least squares based iterative solutions of coupled fuzzy Sylvester matrix equations (1) which can be written as Similarly where Furthermore, it can be concluded that Now, the coupled fuzzy Sylvester matrix equations (1) can be written as is a {1, 3}-inverse of the matrix S, where T {1,3} and Q {1,3} are {1, 3}-inverse of the matrices T and Q, respectively. In particular, the Moore-Penrose inverse of the matrix S is: We now outline our suggested algorithm.

Algorithm III
Step 1 Input crisp matrices A, D ∈R m×m and B, E ∈R n×n and input fuzzy matrices C, F ∈R m×n , and number ε.

Iterative Algorithm for Solving the Coupled Fuzzy Sylvester Matrix Equations (2)
In this section, we introduce an iterative least-squares solution of coupled fuzzy Sylvester matrix equations given in (2), where A, C, E, H ∈R m×m and B, D, G, N ∈ R n×l are given crisp matrices and F 1 , F 2 ∈ R m×l are given fuzzy matrices while X, Y ∈ R m×n are fuzzy matrices to be determined. The basic idea is to regard Equation (2) as two matrices Hence, Equation (2) can be decomposed into two matrix equations of the form: where S 1 = A E and T 1 = B G where S 2 = C, H and T 2 = D, N Then we can define the following iterative formulas where μ is the convergence factor. Substituting from Equations (35) and (36) into Equations (39) and (40) gives The right-hand sides of these equations contain the unknown fuzzy matrices X and Y, so it is impossible to realize the algorithm in Equations (41) and (42). By applying the hierarchical identification principle [4], the unknown fuzzy matrices X and Y in these equations is replaced with its estimate X(k) and Y(k). Thus one has In this case, the iterative least-squares solutions of coupled fuzzy Sylvester matrix equations can be written as We now outline our suggested algorithm.

Algorithm IV
Step 1 Input crisp matrices A, C, E, H ∈R m×m and B, D, G, N ∈R n×l and input fuzzy matrices F 1 , F 2 ∈R m×l , and number ε.
Step 6 Set k = k + 1 and return to step 4.

Theorem 3.2:
If the coupled fuzzy Sylvester matrix equations (2) are consistent and has a unique fuzzy solutions X * = (X * , X * ) ∈R m×n and Y * = (Y * , Y * ) ∈R m×n and then the iterative sequence {X(k)}, {X(k)}, {Y(k)} and {Y(k)} generated by algorithm IV converges to X * , X * , Y * and Y * , that is, lim

Proof:
The proof is similar to Theorem 3.1.

Least Squares Based Iterative Solutions of Coupled Fuzzy Sylvester Matrix Equations (2)
In this subsection, we study least squares based iterative solutions of coupled fuzzy Sylvester matrix equations (2) that can be written as Similarly where In addition, it can be concluded that Now, the coupled fuzzy Sylvester matrix equations (2) can be written as S M = N, where Now, applying Lemma 2.1 for the matrix equation S M = N, then the following least squares based iterative algorithm leads to lim k→∞ M(k) = M ⎡ then the matrix is a {1, 3}-inverse of the matrix S, where T {1,3} and Q {1,3} are {1, 3}-inverse of the matrices T and Q, respectively. In particular, the Moore-Penrose inverse of the matrix S is We now outline our suggested algorithm.

Algorithm V
Step 1 Input crisp matrices A, C, E, H ∈R m×m and B, D, G, N ∈R n×l and input fuzzy matrices F 1 , F 2 ∈R m×l , and number ε.

Numerical Examples
Numerical examples to demonstrate the efficacy of the proposed algorithms are given in this section.

Example 4.1:
In this example, we demonstrate our algorithm I and algorithm II theoretical results for solving coupled fuzzy Sylvester matrix equations given in (1),  This system of coupled fuzzy Sylvester matrix equations has a unique solution Algorithm I and algorithm II are applied to solve generalized Sylvester matrix equations (1).
When selecting the initial matrices as X(0), . Algorithm I is convergent for 0 < μ < 0.5 and the iteration process stops at k = 195. While, algorithm II is convergent for 0 < μ < 0.5 and the iteration process stops at k = 120. The iterative solution X(k) = (X(k), X(k)) and Y(k) = (Y(k), Y(k)) for algorithm I is given in Table 1 for μ = 0.5. And the iterative solution X(k) = (X(k), X(k)) and Y(k) = (Y(k), Y(k)) for algorithm II is given in Table 2 for μ = 0.5. We can see that the suggested modified Iterative algorithm (algorithm II) converges faster than iterative algorithm I to solve the coupled fuzzy Sylvester matrix equations (1)

Example 4.2:
In this example, we demonstrate our algorithm III theoretical results for solving coupled fuzzy Sylvester matrix equations given in (1),  This system of coupled fuzzy Sylvester matrix equations has a unique solution We use algorithm III to solve generalized Sylvester matrix equations (1).
When the initial matrices are chosen as  (1), We apply algorithm V to solve the generalized Sylvester matrix equations (2).

Conclusion
In this paper, five iterative algorithms have been constructed to solve two coupled fuzzy Sylvester matrix equations. Two iterative algorithms are based on the generalized inverse of the coefficient matrix, then iterative solutions are constructed by applying the hierarchical identification principle and by using the block-matrix inner product to solve the two coupled fuzzy Sylvester matrix equations (1) and (2). Also, two least-squares iterative algorithm to solve the two coupled fuzzy Sylvester matrix equations (1) and (2). And a modified iterative algorithm for solving the coupled fuzzy Sylvester matrix equations (1) is proposed. This proposed modification is compared with the first algorithm where our modification exhibits fast convergence behavior. When these two coupled fuzzy Sylvester matrix equations are consistent, for any initial arbitrary fuzzy matrices X(0), Y(0) the solutions can be obtained. We tested the proposed algorithms using MATLAB and the results verify our theoretical findings.