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

Solving Two Coupled Fuzzy Sylvester Matrix Equations Using Iterative Least-squares Solutions FUZZY INFORMATION AND ENGINEERING 2020, VOL. 12, NO. 4, 464–489 https://doi.org/10.1080/16168658.2021.1923442 Solving Two Coupled Fuzzy Sylvester Matrix Equations Using Iterative Least-squares Solutions a b Ahmed M. E. Bayoumi and Mohamed A. Ramadan a b Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt; Department of Mathematics, Faculty of Science, Menoufia University, Shebeen El- Koom, Egypt ABSTRACT ARTICLE HISTORY Received 21 January 2019 In this paper, five iterative methods for solving two coupled fuzzy Revised 10 July 2020 Sylvester matrix equations are considered. The two coupled fuzzy Accepted 26 April 2021 Sylvester matrix equations are expressed by using the generalized inverse of the coefficient matrix, then iterative solutions are con- KEYWORDS structed by applying the hierarchical identification principle and by Coupled fuzzy Sylvester using the block-matrix inner product (the star product for short). A matrix equations; Iterative proposed modification to this algorithm to solve the first coupled algorithm; Kronecker product; Frobenius norm; fuzzy Sylvester matrix equations is suggested. This proposed modifi- Star product cation 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. 1. 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 AX B = F and the generalized Sylvester matrix equation AX B + CX D = F, by extending Jacobi and Gauss–Seidel iteration methods for Ax = b. Ding and Chen [2] suggested a least-squares iterative algorithm to solve the generalized coupled Sylvester matrix equation AX + YB = C, DX + YE = F (1) 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 algo- rithms [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 gen- eralized reflexive solution X and the generalized anti-reflexive solution Y of the coupled CONTACT Ahmed M. E. Bayoumi ame_bayoumi@yahoo.com © 2021 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society of China & Operations Research Society of Guangdong Province. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 465 Sylvester matrix equations AXB + CYD = F , EXG + HYN = F.(2) 1 2 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 col- umn. 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 lin- ear 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, defi- nitions, 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 equa- tions 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 sec- ond 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 prop- erties of these iterative algorithms. In Section4, numerical examples are introduced to illustrate the effectiveness of the proposed algorithms. 2. Preliminaries The following notations, definitions, lemmas and theorems will be used to develop the proposed work. We use A to denote the transpose of A.Thesetofall m × n real matri- m×n m×n T T T T ces is denoted by R .For A ∈ R , vec (A) is defined as vec (A) = [a a ······ a ] 1 2 n where a is the ith column of the matrix A. The Kronecker product of two matrices A = (a ) and B is denoted by A ⊗ B. We have the following well-known property ij m×n vec (MX N) = (N ⊗ M) vec (X) for matrices M, X, N. 466 A. M. E. BAYOUMI AND M. A. RAMADAN Definition 2.1: Block-matrix inner product [2] The block-matrix inner product is called the star product for short, denoted by (∗).Let ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ X Y A A ··· A 1 1 11 12 1p ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ X Y A A ··· A 2 2 21 22 2p ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ mp×n np×m X = ⎢ ⎥ ∈ R , Y = ⎢ ⎥ ∈ R , S = ⎢ ⎥ , . . A . . . . . . . . . ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ . . . . . X Y A A ··· A p p p1 p2 pp ⎡ ⎤ B B ··· B 11 12 1p ⎢ ⎥ B B ··· B 21 22 2p ⎢ ⎥ S = ⎢ ⎥ . B . . . . . . . ⎣ ⎦ . . . B B ··· B p1 p2 pp Then the block-matrix star product is defined as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ X Y Y A X A X ··· A X 1 1 1 11 1 12 2 1p p ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ X Y Y A X A X ··· A X 2 2 2 21 1 22 2 2p p ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ X ∗ Y = ∗ = , S ∗ X = , ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ . . . A . . . . . . . . . . ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ . . . . . . X Y Y A X A X ··· A X p p p p1 1 p2 2 pp p ⎡ ⎤ ⎡ ⎤ X B X B ··· X B A B A B ··· A B 1 11 1 12 1 1p 11 11 12 12 1p 1p ⎢ ⎥ ⎢ ⎥ X B X B ··· X B A B A B ··· A B 2 21 2 22 2 2p 21 21 22 22 2p 2p ⎢ ⎥ ⎢ ⎥ X ∗ S = , S ∗ S = . ⎢ ⎥ ⎢ ⎥ B . . . A B . . . . . . . . . . . . . ⎣ ⎦ ⎣ ⎦ . . . . . . . . X B X B ··· X B A B A B ··· A B p p1 p p2 p pp p1 p1 p2 p2 pp pp The following basic concepts of fuzzy number arithmetic and fuzzy linear system of equa- tions 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: 1) u(r) is a bounded left continuous non-decreasing function over [0, 1], 2) u(r) is a bounded right continuous non-increasing function over [0, 1], 3) u(r) ≤ u(r),0 ≤ r ≤ 1. A crisp number α is simply represented by u(r) = u(r) = α,0 ≤ r ≤ 1. The triangular fuzzy numbers are very popular and denoted by u = (c, α, β) and defined by x − c + α c − α ≤ x ≤ c, ⎨ α c + β − x u(x) = c ≤ x ≤ c + β, 0 otherwise. where α> 0and β> 0. The parametric form of the number is u(r) = rα + c − α, u(r) = c + β − β r. FUZZY INFORMATION AND ENGINEERING 467 The addition and scalar multiplication of fuzzy numbers are defined by the extension principle and can be equivalently represented as follows, see [8, 9]. For arbitrary fuzzy numbers v = (v (r), v (r)) and w = (w (r), w (r)),0 ≤ r ≤ 1and real number k as follows: a) v = w if and only if v (r) = w (r) and v (r) = w (r), b) v + w = ( v (r) + w (r), v (r) + w (r)), c) v − w = ( v (r) − w (r), v (r) − w (r)), (kv (r), k v (r)) k ≥ 0, d) kv = ( k v (r) , kv (r)) k < 0. Definition 2.3: Consider the p × q linear system of equations a v + a v + ··· + a v = w , ⎪ 11 1 12 2 1q q 1 a v + a v + ··· + a v = w , 21 1 22 2 2n q 2 (3) ⎪ . a v + a v + ··· + a v = w , p1 1 p2 2 pq q p p×q T where the coefficient matrix A = (a ) ∈R is given crisp matrix and w = (w , w , ... , w ) ij 1 2 p is given vector of fuzzy numbers and v = (v , v , ... , v ) is vector of fuzzy numbers to be 1 2 q determined. This system is called an FSLE. Definition 2.4: A fuzzy number vector v = (v , v , ... , v ) where v = (v (r), v (r)),0 ≤ 1 2 q i i r ≤ 1, i = 1, 2, ... , q, is called a solution of the fuzzy linear system of equations (3) if q q ⎪ a v = a v = w , ij j ij j i j=1 j=1 (4) ⎪ q q ⎪ a v = a v = w ij j ij j i j=1 j=1 In general, an arbitrary equation for either w or w is a linear combination of v ’s and v ’s, i i j j respectively. Therefore, in order to solve Equation (3) one must solve a 2p × 2q crisp linear system of equations (5) as follows: Sv = w (5) where S S v w 1 2 S = , v = , w =.(6) S S v w 2 1 where the element of S = (s ),1 ≤ i, j ≤ 2q, as follows: ij if a ≥ 0 ⇒ s = a , s = a ij i,j i,j i+p,j+q i,j (7) if a < 0 ⇒ s =−a , s =−a ij i,j+q ij i+p,j i,j 468 A. M. E. BAYOUMI AND M. A. RAMADAN Theorem 2.1: [10]: Let matrix S be in the form (6), then the matrix {1,3} {1,3} {1,3} {1,3} 1 (S + S ) + (S − S ) (S + S ) − (S − S ) 1 2 1 2 1 2 1 2 {1,3} S = (8) {1,3} {1,3} {1,3} {1,3} (S + S ) − (S − S ) (S + S ) + (S − S ) 1 2 1 2 1 2 1 2 {1,3} {1,3} is a {1, 3}-inverse of the matrix S, where (S + S ) and (S − S ) are {1, 3}-inverse of 1 2 1 2 the matrices (S + S ) and (S − S ), respectively. In particular, the Moore–Penrose inverse 1 2 1 2 of the matrix S is † † † † (S + S ) + (S − S ) (S + S ) − (S − S ) 1 2 1 2 1 2 1 2 S = .(9) † † † † 2 (S + S ) − (S − S ) (S + S ) + (S − S ) 1 2 1 2 1 2 1 2 Theorem 2.2: [10]: {1,3} For the consistent system (5) and any {1, 3}-inverse S of the coefficient matrix S, {1,3} v = S 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 x(k) = x k→∞ T −1 T x(k) = x(k − 1) + μ(A A) A [b − Ax(k − 1)], 0 <μ< 2. 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): T −1 T T T −1 X(k) = X(k − 1) + μ(A A) A [F − AX(k − 1)B]B (BB ) ,0 <μ< 2. Lemma 2.3: [2] The coupled fuzzy Sylvester matrix equations given in (1), AX + YB = C, DX + YE = F, m×m n×n m×n where A, D ∈R and B, E ∈R are given crisp matrices and C, F ∈R are given fuzzy m×n matrices while X, Y ∈R are fuzzy matrices to be determined. Equation (1) has a unique solution if and only if the matrix I ⊗AB ⊗ I n m 2mn×2mn Q = ∈ R I ⊗DE ⊗ I n m is non-singular; in this case, the unique solution is given by vec(X) vec(C) −1 = Q vec(Y) vec(F) and the corresponding homogeneous matrix equation AX + YB = 0, DX + YE = 0has a unique solution X = Y = 0. FUZZY INFORMATION AND ENGINEERING 469 Lemma 2.4: The coupled Sylvester matrix equations given in (2), AXB + CYD = F , EXG + HYN = F . 1 2 m×m n×l m×l where A, C, E, H ∈R and B, D, G, N ∈R are given crisp matrices and F , F ∈R are 1 2 m×n given fuzzy matrices while X, Y ∈R are fuzzy matrices to be determined. Equation (2) has a unique solution if and only if the matrix T T B ⊗AD ⊗ C 2lm×2nm Q = ∈ R T T G ⊗EN ⊗ H is non-singular; in this case, the unique solution is given by vec(X) vec(F ) −1 = Q vec(Y) vec(F ) and the corresponding homogeneous matrix equation AXB + CYD = 0, EXG + HYN = 0 has a unique solution X = Y = 0. 3. 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]. 3.1. 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), AX + YB = C, DX + YE = F m×m n×n m×n where A, D ∈R and B, E ∈R are given crisp matrices and C, F ∈ R are given fuzzy m×n matrices while X, Y ∈ R are fuzzy matrices to be determined. The basic idea is to regard Equation (1) as two matrices C − YB R = (10) F − YE R = C − AX, F − DX (11) Hence, Equation (1) can be decomposed into two matrix equations of the form: K X = R (12) 1 1 YK = R (13) 2 2 Here K = and K = B, E 1 2 Then, we can define the following iterative formulas T −1 T X(k) = X(k − 1) + μ(K K ) K (R − K X(k − 1)) (14) 1 1 1 1 1 470 A. M. E. BAYOUMI AND M. A. RAMADAN T T −1 Y(k) = Y(k − 1) + μ(R − Y(k − 1) K ) K (K K ) (15) 2 2 2 2 2 where μ is the convergence factor. Substituting from Equations (10) and (11) into Equa- tions (14) and (15) gives A C − AX(k − 1) − YB T −1 X(k) = X(k − 1) + μ(K K ) , (16) D F − DX(k − 1) − YE T −1 Y(k) = Y(k − 1) + μ C − AX − Y(k − 1)B, F − DX − Y(k − 1)E B, E (K K ) (17) 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 hierar- chical 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 A C − AX(k − 1) − Y(k − 1)B T −1 X(k) = X(k − 1) + μ(K K ) , (18) D F − DX(k − 1) − Y(k − 1)E Y(k) = Y(k − 1) + μ C − AX(k − 1) − Y(k − 1)B, F − DX(k − 1) − Y(k − 1)E T −1 × B, E (K K ) (19) 2 2 μ = or μ = (20) −1 −1 m + n T T T T λ (K (K K ) K ) + λ (K (K K ) K ) max 1 1 max 2 2 1 1 2 2 In this case, the iterative least-squares solutions of coupled fuzzy Sylvester matrix equa- tions can be written as A C − AX(k − 1) − Y(k − 1)B T −1 X(k) = X(k − 1) + μ(K K ) , (21) D F − DX(k − 1) − Y(k − 1)E A C − AX(k − 1) − Y(k − 1)B T −1 X(k) = X(k − 1) + μ(K K ) , (22) D F − DX(k − 1) − Y(k − 1)E Y(k) = Y(k − 1) + μ C − AX(k − 1) − Y(k − 1)B, F − DX(k − 1) − Y(k − 1)E T −1 × B, E (K K ) (23) Y(k) = Y(k − 1) + μ C − AX(k − 1) − Y(k − 1)B, F − DX(k − 1) − Y(k − 1)E T −1 × B, E (K K ) (24) 2 2 μ = or μ = −1 −1 T T T T m + n λ (K (K K ) K ) + λ (K (K K ) K ) max 1 1 max 2 2 1 1 2 2 We now outline our suggested algorithm. Algorithm I m×m n×n m×n Step 1 Input crisp matrices A, D ∈R and B, E ∈R and input fuzzy matrices C, F ∈R , and number ε. FUZZY INFORMATION AND ENGINEERING 471 m×n Step 2 Given any two initial fuzzy matrices X(0), Y(0) ∈R . Step 3 Compute K = and K = B, E 1 2 Step 4 For k = 1, 2, ··· until convergence A C − AX(k − 1) − Y(k − 1)B T −1 X(k) = X(k − 1) + μ(K K ) , D F − DX(k − 1) − Y(k − 1)E A C − AX(k − 1) − Y(k − 1)B T −1 X(k) = X(k − 1) + μ(K K ) , D F − DX(k − 1) − Y(k − 1)E Y(k) = Y(k − 1) + μ C − AX(k − 1) − Y(k − 1)B, F − DX(k − 1) − Y(k − 1)E T −1 × B, E (K K ) Y(k) = Y(k − 1) + μ C − AX(k − 1) − Y(k − 1)B, F − DX(k − 1) − Y(k − 1)E T −1 × (K K ) B, E 2 2 μ = or μ = −1 −1 T T T T m + n λ (K (K K ) K ) + λ (K (K K ) K ) max 1 1 max 2 2 1 1 2 2 Step 5 If ||X(k) − X(k-1) ||/||X(k)|| <ε, ||X(k) − X(k-1) ||/||X(k)|| <ε, ||Y(k) − Y(k-1) ||/||Y(k)|| <ε and ||Y(k) − Y(k-1) ||/||Y(k)|| <ε stop; otherwise go to step 6. Step 6 Set k = k + 1 and return to step 4. Step 7 End. Theorem 3.1: If the coupled fuzzy Sylvester matrix equations (1) is consistent and has a ∗ ∗ m×n ∗ ∗ m×n ∗ ∗ unique fuzzy solutions X = (X , X ) ∈R and Y = (Y , Y ) ∈R and 2 2 μ = or μ = −1 −1 T T T T m + n λ (K (K K ) K ) + λ (K (K K ) K ) max 1 1 max 2 2 1 1 2 2 then the iterative sequence {X(k)}, {X(k)}, {Y(k)} and {Y(k)} generated by algorithm I con- ∗ ∗ ∗ ∗ ∗ ∗ ∗ verges to X , X , Y and Y , that is, lim X(k) = X ,lim X(k) = X ,lim Y(k) = Y and k→∞ k→∞ k→∞ lim Y(k) = Y for any initial fuzzy matrices X(0) , X(0) , Y(0) and Y(0) . k→∞ Proof: First, we define the estimation error matrices as ∗ ∗ ∗ ∗ ξ (k) = X(k) − X , ξ (k) = Y(k) − Y , ξ (k) = X(k) − X and ξ (k) = Y(k) − Y for 1 2 3 4 k = 1, 2, ··· . Using algorithm I and the above error matrices, we can obtain A Aξ (k − 1) + ξ (k − 1)B 1 2 T −1 ξ (k) = ξ (k − 1) − μ(K K ) (25) 1 1 1 D Dξ (k − 1) + ξ (k − 1)E 1 2 ξ (k) = ξ (k − 1) − μ Aξ (k − 1) + ξ (k − 1)B, Dξ (k − 1) + ξ (k − 1)E 2 2 1 2 1 2 T −1 × (K K ) (26) B, E 2 472 A. M. E. BAYOUMI AND M. A. RAMADAN Now, by taking the norm of (25) and (26) and using the following formula, we have −1 −1 −1 T 2 T T T ||K [X + (K K ) Y ] || = tr{[X + (K K ) Y] (K K )[X + (K K ) Y] } 1 1 1 1 1 1 1 1 1 −1 T T T T T = tr {X (K K ) X + 2 X Y + Y (K K ) Y } 1 1 1 1 −1 2 T T =||K X|| + 2 tr[ X Y ] + (K K ) Y . (27) 1 1 Gives 2 T T ||K ξ (k)|| = tr{ξ (k) K K ξ (k)} 1 1 1 1 1 1 T T = tr{ξ (k − 1) K K ξ (k − 1)} 1 1 1 1 A Aξ (k − 1) + ξ (k − 1)B 1 2 − 2 μ tr ξ (k − 1) D Dξ (k − 1) + ξ (k − 1)E 1 2 −1 Aξ (k − 1) + ξ (k − 1)B 1 2 2 T T + μ (K K ) K 1 1 Dξ (k − 1) + ξ (k − 1)E 1 2 2 T ≤||K ξ (k − 1)|| − 2 μ tr{[Aξ (k − 1)] (Aξ (k − 1) + ξ (k − 1)B) 1 1 1 1 2 + [Dξ (k − 1)] (Dξ (k − 1) + ξ (k − 1)E)} 1 1 2 2 2 2 + μ m [||Aξ (k − 1) + ξ (k − 1)B|| +||Dξ (k − 1) + ξ (k − 1)E|| ] (28) 1 2 1 2 Similarly, 2 T T ||ξ (k) K || = tr{ξ (k) K K ξ (k)} 2 2 2 2 2 2 T T = tr{ξ (k − 1) K K ξ (k − 1)} 2 2 2 2 − 2 μ tr Aξ (k − 1) + ξ (k − 1)B, Dξ (k − 1) + ξ (k − 1)E 1 2 1 2 × B, E ξ (k − 1) + μ Aξ (k − 1) + ξ (k − 1)B, Dξ (k − 1) + ξ (k − 1)E 1 2 1 2 T −1 × B, E (K K ) 2 T ≤||ξ (k − 1) K || − 2 μ tr{[ξ (k − 1)B] (Aξ (k − 1) 2 2 2 1 + ξ (k − 1)B) + [ξ (k − 1)E] (Dξ (k − 1) + ξ (k − 1)E)} 2 2 1 2 2 2 2 + μ n [||Aξ (k − 1) + ξ (k − 1)B|| +||Dξ (k − 1) + ξ (k − 1)E|| ] (29) 1 2 1 2 Define the nonnegative definite function η(k) by 2 2 η(k) =||K ξ (k)|| +||ξ (k) K || . 1 1 2 2 FUZZY INFORMATION AND ENGINEERING 473 From (28) and (29), this function can be computed as 2 T η(k) ≤||K ξ (k − 1)|| − 2 μ tr{[Aξ (k − 1)] (Aξ (k − 1) + ξ (k − 1)B) 1 1 1 1 2 + [Dξ (k − 1)] ( Dξ (k − 1) + ξ (k − 1)E)} 1 1 2 2 2 2 + μ m [||Aξ (k − 1) + ξ (k − 1)B|| +||Dξ (k − 1) + ξ (k − 1)E|| ] 1 2 1 2 2 T +||ξ (k − 1) K || − 2 μ tr{[ξ (k − 1)B] (Aξ (k − 1) + ξ (k − 1)B) 2 2 2 1 2 + [ξ (k − 1)E] (Dξ (k − 1) + ξ (k − 1)E)} 2 1 2 2 2 + μ n [||Aξ (k − 1) + ξ (k − 1)B|| +||Dξ (k − 1) + ξ (k − 1)E|| ] 1 2 1 2 ≤||K ξ (k − 1)|| 1 1 2 T +||ξ (k − 1) K || − 2 μ tr{(Aξ (k − 1) + ξ (k − 1)B) (Aξ (k − 1) + ξ (k − 1)B) 2 2 1 2 1 2 +(Dξ (k − 1) + ξ (k − 1)E) (Dξ (k − 1) + ξ (k − 1)E)} 1 2 1 2 2 2 2 + μ (m + n) [||Aξ (k − 1) + ξ (k − 1)B|| +||Dξ (k − 1) + ξ (k − 1)E|| ] 1 2 1 2 ≤ η(k − 1) − 2 μ[||Aξ (k − 1) + ξ (k − 1)B|| 1 2 +||Dξ (k − 1) + ξ (k − 1)E|| ] 1 2 2 2 2 + μ (m + n) [||Aξ (k − 1) + ξ (k − 1)B|| +||Dξ (k − 1) + ξ (k − 1)E|| ] 1 2 1 2 ≤ η(k − 1) − μ [2 − μ(m + n)][||Aξ (k − 1) + ξ (k − 1)B || 1 2 +||Dξ (k − 1) + ξ (k − 1)E || ] 1 2 k−1 2 2 η(k) ≤ η(1) − μ [2 − μ(m + n) ] [||Aξ (i) + ξ (i)B || +||Dξ (i) + ξ (i)E || ]. 1 2 1 2 i=1 If the convergence factor μ is chosen to satisfy 0 <μ< . m + n Then 2 2 [ ||Aξ (k) + ξ (k)B|| +||Dξ (k) + ξ (k)E|| ] < ∞ 1 2 1 2 k=1 Since the matrix equation (1) has a unique fuzzy solution it follows that as k →∞ lim Aξ (k) + ξ (k)B = 0 and lim Dξ (k) + ξ (k)E = 0. 1 2 1 2 k→∞ k→∞ According to lemma 2.3, we have lim ξ (k) =0and lim ξ (k) = 0. 1 2 k→∞ k→∞ Or ∗ ∗ lim X(k) = X and lim Y(k) = Y k→∞ k→∞ 474 A. M. E. BAYOUMI AND M. A. RAMADAN Similarly, we can prove that ∗ ∗ lim X(k) = X and lim Y(k) = Y . k→∞ k→∞ 3.2. 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), AX + YB = C, DX + YE = F. m×m n×n m×n where A, D ∈R and B, E ∈R are given crisp matrices and C, F ∈R are given fuzzy m×n matrices while X, Y ∈R are fuzzy matrices to be determined. The proposed algorithm is as follows: m×m n×n Algorithm 1: Step 1 Input crisp matrices A, D ∈R and B, E ∈R and input fuzzy m×n matrices C, F ∈R , and number ε. m×n Step 2 Given any two initial fuzzy matrices X(0), Y(0) ∈R . Step 3 Compute K = and K = B, E 1 2 Step 4 For k = 1, 2, ··· until convergence A C − AX(k − 1) − Y(k − 1)B T −1 X(k) = X(k − 1) + μ(K K ) , D F − DX(k − 1) − Y(k − 1)E Y(k) = Y(k − 1) + μ C − AX(k) − Y(k − 1)B, F − DX(k) − Y(k − 1)E T −1 × B, E (K K ) , A C − AX(k − 1) − Y(k − 1)B T −1 X(k) = X(k − 1) + μ(K K ) , D F − DX(k − 1) − Y(k − 1)E Y(k) = Y(k − 1) + μ C − AX(k) − Y(k − 1)B, F − DX(k) − Y(k − 1)E T −1 × B, E (K K ) , 2 2 μ = or μ = −1 −1 T T T T m + n λ (K (K K ) K ) + λ (K (K K ) K ) max 1 1 max 2 2 1 1 2 2 Step 5 If ||X(k) − X(k-1) ||/||X(k)|| <ε, ||X(k) − X(k-1) ||/||X(k)|| <ε, ||Y(k) − Y(k-1) ||/||Y(k)|| <ε and ||Y(k) − Y(k-1) ||/||Y(k)|| <ε stop; otherwise go to step 6. Step 6 Set k = k + 1 and return to step 4. 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). FUZZY INFORMATION AND ENGINEERING 475 3.3. 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 I ⊗AB ⊗ I vec (X) vec (C) n m I ⊗DE ⊗ I vec (Y) vec (F) n m + + + + − − − − If A , B , D , E contain the positive entries of A, B, D, E, respectively, and A , B , D , E contain the absolute value of negative entries of A, B, D, E, respectively, it is obvious that + − + − + − + − A = A − A , B = B − B , D = D − D , E = E − E . So, according to the properties of Kronecker operators it can be written as + − + − I ⊗ A = I ⊗ (A − A ) = I ⊗ A − I ⊗ A , n n n n T T T + − T + − and B ⊗ I = (B − B ) ⊗ I = B ⊗ I − B ⊗ I . m m m m Similarly + − I ⊗ D = I ⊗ D − I ⊗ D , n n n and T T T + − E ⊗ I = E ⊗ I − E ⊗ I . m m m I ⊗AB ⊗ I n m Q = I ⊗DE ⊗ I n m T T + − + − I ⊗ A − I ⊗ A B ⊗ I − B ⊗ I n n m m Q = T T + − + − I ⊗ D − I ⊗ D E ⊗ I − E ⊗ I n n m m Q = S − S (30) 1 2 where T T + + − − I ⊗ A B ⊗ I I ⊗ A B ⊗ I n m n m S = , S = . 1 2 T T + + − − I ⊗ D E ⊗ I I ⊗ D E ⊗ I n m n m Furthermore, it can be concluded that T T + + − − I ⊗ A B ⊗ I I ⊗ A B ⊗ I n m n m T = + T T + + − − I ⊗ D E ⊗ I I ⊗ D E ⊗ I n m n m T = S + S (31) 1 2 Now, the coupled fuzzy Sylvester matrix equations (1) can be written as SM = N (32) T T + + − − S S I ⊗ A B ⊗ I I ⊗ A B ⊗ I 1 2 n m n m S = , S = , S = 1 2 T T + + − − S S 2 1 I ⊗ D E ⊗ I I ⊗ D E ⊗ I n m n m ⎡ ⎤ ⎡ ⎤ vec(X) vec(C) where , ⎢ ⎥ ⎢ ⎥ vec(Y) vec(F) ⎢ ⎥ ⎢ ⎥ M = , N = ⎣ ⎦ ⎣ ⎦ − vec(X) − vec(C) − vec(Y) − vec(F) 476 A. M. E. BAYOUMI AND M. A. RAMADAN By using Lemma 2.1 for the matrix equation (32), then the following least squares based iterative algorithm leads to lim M(k) = M k→∞ ⎡ ⎤ ⎡ ⎤ vec(X(k)) vec(X(k-1)) ⎢ ⎥ ⎢ ⎥ vec(Y(k)) vec(Y(k-1)) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ − vec(X(k)) − vec(X(k-1)) ⎣ ⎦ ⎣ ⎦ − vec(Y(k)) − vec(Y(k-1)) ⎛ ⎡ ⎤ vec(C) vec(X(k-1)) ⎜ ⎢ ⎥ ⎟ ⎢ ⎥ vec(F) S S vec(Y(k-1)) ⎜ ⎢ ⎥ ⎟ T −1 T 1 2 ⎢ ⎥ + μ(S S) S ⎜ − ⎢ ⎥ ⎟ , ⎣ ⎦ ⎝ − vec(C) S S − vec(X(k-1))⎦ ⎠ 2 1 − vec(F) − vec(Y(k-1)) 0 <μ< 2 + + S S I ⊗ A B ⊗ I 1 2 n m Corollary 3.1: Let matrix S be in the form S = where S = , + + S S 2 1 I ⊗ D E ⊗ I n m − − I ⊗ A B ⊗ I n m S = , then the matrix − − I ⊗ D E ⊗ I n m {1,3} {1,3} {1,3} {1,3} T + Q T − Q {1,3} S = (33) {1,3} {1,3} {1,3} {1,3} T − Q T + Q {1,3} {1,3} is a {1, 3}-inverse of the matrix S, where T and Q are {1, 3}-inverse of the matrices T and Q, respectively. In particular, the Moore–Penrose inverse of the matrix S is: † † † † T + Q T − Q S = (34) † † † † 2 T − Q T + Q We now outline our suggested algorithm. Algorithm III m×m n×n Step 1 Input crisp matrices A, D ∈R and B, E ∈R and input fuzzy matrices m×n C, F ∈R , and number ε. m×n Step 2 Given any two initial fuzzy matrices X(0), Y(0) ∈R . Step 3 Compute T T + + − − I ⊗ A B ⊗ I I ⊗ A B ⊗ I S S n m n m 1 2 S = , S = , S = . 1 2 T T + + − − S S I ⊗ D E ⊗ I I ⊗ D E ⊗ I 2 1 n m n m + + + + where A , B , D , E contain the positive entries of A, B, D, E, respectively, and − − − − A , B , D , E contain the absolute value of negative entries of A, B, D, E, respectively, + − + − + − + − where A = A − A , B = B − B , D = D − D , E = E − E . FUZZY INFORMATION AND ENGINEERING 477 Step 4 For k = 1, 2, ··· until convergence ⎡ ⎤ ⎡ ⎤ vec(X(k)) vec(X(k-1)) ⎢ ⎥ ⎢ ⎥ vec(Y(k)) vec(Y(k-1)) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ − vec(X(k)) − vec(X(k-1)) ⎣ ⎦ ⎣ ⎦ − vec(Y(k)) − vec(Y(k-1)) ⎛ ⎡ ⎤ ⎞ ⎡ ⎤ vec(C) vec(X(k-1)) ⎜ ⎢ ⎥ ⎟ ⎢ ⎥ vec(F) S S vec(Y(k-1)) ⎜ ⎢ ⎥ ⎟ 1 2 T −1 T ⎢ ⎥ + μ(S S) S ⎜ − ⎢ ⎥ ⎟ , ⎣ ⎦ − vec(C) S S − vec(X(k-1)) ⎝ ⎣ ⎦ ⎠ 2 1 − vec(F) − vec(Y(k-1)) 0 <μ< 2 Step 5 If ||X(k) − X(k-1) ||/||X(k)|| <ε, ||X(k) − X(k-1) ||/||X(k)|| <ε, ||Y(k) − Y(k-1) ||/||Y(k)|| <ε and ||Y(k) − Y(k-1) ||/||Y(k)|| <ε stop; otherwise go to step 6. Step 6 Set k = k + 1 and return to step 4. Step 7 End. 3.4. 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), AXB + CYD = F , EXG + HYN = F . 1 2 m×m n×l m×l where A, C, E, H ∈R and B, D, G, N ∈ R are given crisp matrices and F , F ∈ R are 1 2 m×n given fuzzy matrices while X, Y ∈ R are fuzzy matrices to be determined. The basic idea is to regard Equation (2) as two matrices F − CYD R = (35) F − HYN R = F − AXB, F − EXG (36) 2 1 2 Hence, Equation (2) can be decomposed into two matrix equations of the form: S X ∗ T1 = R (37) 1 1 S ∗ YT = R (38) 2 2 2 A B where S = and T = 1 1 E G where S = C, H and T = D, N 2 2 Then we can define the following iterative formulas T −1 T T T −1 X(k) = X(k − 1) + μ(S S ) S [R − S X(k − 1) ∗ T1] ∗ T ( T T ) (39) 1 1 1 1 1 1 1 1 T −1 T T T −1 Y(k) = Y(k − 1) + μ(S S ) S ∗ [R − S ∗ Y(k − 1)T ] T ( T T ) (40) 2 2 2 2 2 2 2 2 2 478 A. M. E. BAYOUMI AND M. A. RAMADAN where μ is the convergence factor. Substituting from Equations (35) and (36) into Equa- tions (39) and (40) gives T T A F − AX(k − 1)B − CYD B T −1 T −1 X(k) = X(k − 1) + μ(S S ) ∗ ( T T ) 1 1 1 1 E F − EX(k − 1)G − HYN G (41) T −1 T Y(k) = Y(k − 1) + μ(S S ) {[C, H] ∗ [F − AXB − CY(k − 1)D, 2 1 T −1 F − EXG − HY(k − 1)N]} D, N ( T T ) (42) 2 2 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 hierar- chical 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 T T A F − AX(k − 1)B − CY(k − 1)D B T −1 X(k) = X(k − 1) + μ(S S ) ∗ E F − EX(k − 1)G − HY(k − 1)N G T −1 × ( T T ) (43) T −1 Y(k) = Y(k − 1) + μ(S S ) × C, H ∗ F − AX(k − 1)B − CY(k − 1)D,F − EX(k − 1)G − HY(k − 1)N 1 2 T −1 × ( T T ) , (44) D, N μ = or m + n μ = −1 −1 −1 −1 T T T T T T T T λ (S (S S ) S )λ (T (T T ) T ) + λ (S (S S ) S )λ (T (T T ) T ) max 1 1 max 1 1 max 2 2 max 2 2 1 1 1 1 2 2 2 2 (45) In this case, the iterative least-squares solutions of coupled fuzzy Sylvester matrix equations canbewrittenas T T A F − AX(k − 1)B − CY(k − 1)D B T −1 X(k) = X(k − 1) + μ(S S ) ∗ E F − EX(k − 1)G − HY(k − 1)N G T −1 T ) , (46) × ( T T T A F − AX(k − 1)B − CY(k − 1)D B T −1 X(k) = X(k − 1) + μ(S S ) ∗ E F − EX(k − 1)G − HY(k − 1)N G T −1 × ( T T ) , (47) T −1 Y(k) = Y(k − 1) + μ(S S ) C, H ∗ F − AX(k − 1)B − CY(k − 1)D, F − EX(k − 1)G − HY(k − 1)N 1 2 T −1 × ( T T ) , (48) D, N T −1 Y(k) = Y(k − 1) + μ(S S ) 2 FUZZY INFORMATION AND ENGINEERING 479 C, H ∗ F − AX(k − 1)B − CY(k − 1)D, F − EX(k − 1)G − HY(k − 1)N 1 2 T −1 × D, N ( T T ) , (49) μ = or m + n μ = −1 −1 −1 −1 T T T T T T T T λ (S (S S ) S )λ (T (T T ) T ) + λ (S (S S ) S )λ (T (T T ) T ) max 1 1 max 1 1 max 2 2 max 2 2 1 1 1 1 2 2 2 2 We now outline our suggested algorithm. Algorithm IV m×m n×l Step 1 Input crisp matrices A, C, E, H ∈R and B, D, G, N ∈R and input fuzzy matrices m×l F , F ∈R , and number ε. 1 2 m×n Step 2 Given any two initial fuzzy matrices X(0), Y(0) ∈R . Step 3 Compute A B S = , T = , S = C, H , andT = D, N 1 1 2 2 E G Step 4 For k = 1, 2, ··· until convergence T T A F − AX(k − 1)B − CY(k − 1)D B T −1 X(k) = X(k − 1) + μ(S S ) ∗ E F − EX(k − 1)G − HY(k − 1)N G T −1 × ( T T ) , T T A F − AX(k − 1)B − CY(k − 1)D B T −1 X(k) = X(k − 1) + μ(S S ) ∗ E F − EX(k − 1)G − HY(k − 1)N G T −1 × ( T T ) , T −1 Y(k) = Y(k − 1) + μ(S S ) × C, H ∗ F − AX(k − 1)B − CY(k − 1)D, F − EX(k − 1)G − HY(k − 1)N 1 2 T −1 × D, N ( T T ) , T −1 Y(k) = Y(k − 1) + μ(S S ) × C, H ∗ F − AX(k − 1)B − CY(k − 1)D,F − EX(k − 1)G − HY(k − 1)N 1 2 T −1 × D, N ( T T ) , μ = or m + n μ = −1 −1 −1 −1 T T T T T T T T λ (S (S S ) S )λ (T (T T ) T ) + λ (S (S S ) S )λ (T (T T ) T ) max 1 1 max 1 1 max 2 2 max 2 2 1 1 1 1 2 2 2 2 480 A. M. E. BAYOUMI AND M. A. RAMADAN Step 5 If ||X(k) − X(k-1) ||/||X(k)|| <ε, ||X(k) − X(k-1) ||/||X(k)|| <ε, ||Y(k) − Y(k-1) ||/||Y(k)|| <ε and ||Y(k) − Y(k-1) ||/||Y(k)|| <ε stop; otherwise go to step 6. Step 6 Set k = k + 1 and return to step 4. Step 7 End. Theorem 3.2: If the coupled fuzzy Sylvester matrix equations (2) are consistent and has a ∗ ∗ m×n ∗ ∗ m×n ∗ ∗ unique fuzzy solutions X = (X , X ) ∈R and Y = (Y , Y ) ∈R and μ = or m + n μ = −1 −1 −1 −1 T T T T T T T T λ (S (S S ) S )λ (T (T T ) T ) + λ (S (S S ) S )λ (T (T T ) T ) max 1 1 max 1 1 max 2 2 max 2 2 1 1 1 1 2 2 2 2 then the iterative sequence {X(k)}, {X(k)}, {Y(k)} and {Y(k)} generated by algorithm IV con- ∗ ∗ ∗ ∗ ∗ ∗ ∗ verges to X , X , Y and Y , that is, lim X(k) = X ,lim X(k) = X ,lim Y(k) = Y and k→∞ k→∞ k→∞ lim Y(k) = Y for any initial fuzzy matrices X(0) , X(0) , Y(0) and Y(0) . k→∞ Proof: The proof is similar to Theorem 3.1. 3.5. 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 T T B ⊗AD ⊗ C vec (X) vec (F ) T T G ⊗EN ⊗ H vec (Y) vec (F ) + + + + + + + + If A , B , C , D , E , G , H , N contain the positive entries of A, B, C, D, E, G, H, N,respec- − − − − − − − − tively, and A , B , C , D , E , G , H , N contain the absolute value of negative entries of + − + − + − A, B, C, D, E, G, H, N,respectively,itisobviousthat A = A − A , B = B − B , C = C − C , + − + − + − + − + − D = D − D , E = E − E , G = G − G , H = H − H , N = N − N . So, according to the properties of Kronecker operators it can be written as T + − T + − B ⊗ A = (B − B ) ⊗ (A − A ) T T T T + + − − + − − + = (B ⊗ A + B ⊗ A ) − (B ⊗ A + B ⊗ A ) , Similarly T T T T T + + − − + − − + D ⊗ C = (D ⊗ C + D ⊗ C ) − (D ⊗ C + D ⊗ C ) , T T T T T + + − − + − − + G ⊗ E = (G ⊗ E + G ⊗ E ) − (G ⊗ E + G ⊗ E ) , FUZZY INFORMATION AND ENGINEERING 481 T T T T T + + − − + − − + and N ⊗ H = (N ⊗ H + N ⊗ H ) − (N ⊗ H + N ⊗ H ) . T T B ⊗AD ⊗ C Q = T T G ⊗EN ⊗ H T T T T + + − − + + − − B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C Q = T T T T + + − − + + − − G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H T T T T + − − + + − − + B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C T T T T + − − + + − − + G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H Q = S − S (50) 1 2 where T T T T + + − − + + − − B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C S = , T T T T + + − − + + − − G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H T T T T + − − + + − − + B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C S = . T T T T + − − + + − − + G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H In addition, it can be concluded that T T T T + + − − + + − − B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C T = T T T T + + − − + + − − G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H T T T T + − − + + − − + B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C T T T T + − − + + − − + G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H T = S + S . (51) 1 2 Now, the coupled fuzzy Sylvester matrix equations (2) can be written as SM = N, where T T T T + + − − + + − − S S B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C 1 2 S = , S = , T T T T + + − − + + − − S S 2 1 G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H ⎡ ⎤ vec(X) T T T T + − − + + − − + ⎢ ⎥ B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C vec(Y) ⎢ ⎥ S = , M = , T T T T + − − + + − − + ⎣ ⎦ − vec(X) G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H − vec(Y) ⎡ ⎤ vec(F ) vec(F ) N = ⎣ ⎦ − vec(F ) − vec(F ) 2 482 A. M. E. BAYOUMI AND M. A. RAMADAN Now, applying Lemma 2.1 for the matrix equation SM = N, then the following least squares based iterative algorithm leads to lim M(k) = M k→∞ ⎡ ⎤ ⎡ ⎤ vec(X(k)) vec(X(k-1)) ⎢ ⎥ ⎢ ⎥ vec(Y(k)) vec(Y(k-1)) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎣ − vec(X(k))⎦ ⎣ − vec(X(k-1))⎦ − vec(Y(k)) − vec(Y(k-1)) ⎛ ⎡ ⎤ ⎞ ⎡ ⎤ vec(X(k-1)) vec(F ) ⎜ ⎢ ⎥ ⎟ ⎢ ⎥ vec(Y(k-1)) ⎜ vec(F ) S S ⎢ ⎥ ⎟ 2 1 2 T −1 T ⎢ ⎥ + μ(S S) S ⎜ − ⎢ ⎥ ⎟ , ⎣ ⎦ − vec(F ) S S ⎝ ⎣ − vec(X(k-1))⎦ ⎠ 1 2 1 − vec(F ) − vec(Y(k-1)) 0 <μ< 2 S S 1 2 Corollary 3.2: Let matrix S be in the form S = where S S 2 1 T T T T + + − − + + − − B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C S = , T T T T + + − − + + − − G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H T T T T + − − + + − − + B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C S = , T T T T + − − + + − − + G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H then the matrix {1,3} {1,3} {1,3} {1,3} T + Q T − Q {1,3} S = (52) {1,3} {1,3} {1,3} {1,3} 2 T − Q T + Q {1,3} {1,3} is a {1, 3}-inverse of the matrix S, where T and Q are {1, 3}-inverse of the matrices T and Q, respectively. In particular, the Moore–Penrose inverse of the matrix S is † † † † 1 T + Q T − Q S = (53) † † † † T − Q T + Q We now outline our suggested algorithm. Algorithm V m×m n×l Step 1 Input crisp matrices A, C, E, H ∈R and B, D, G, N ∈R and input fuzzy matrices m×l F , F ∈R , and number ε. 1 2 m×n Step 2 Given any two initial fuzzy matrices X(0), Y(0) ∈R . Step 3 Compute T T T T + + − − + + − − S S B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C 1 2 S = , S = , T T T T + + − − + + − − S S 2 1 G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H T T T T + − − + + − − + B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C S = T T T T + − − + + − − + G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H FUZZY INFORMATION AND ENGINEERING 483 + + + + + + + + where A , B , C , D , E , G , H , N contain the positive entries of A, B, C, D, E, G, H, N, − − − − − − − − respectively, and A , B , C , D , E , G , H , N contain the absolute value of negative + − + − + − + entries of A, B, C, D, E, G, H, N where A = A − A , B = B − B , C = C − C , D = D − − + − + − + − + − D , E = E − E , G = G − G , H = H − H , N = N − N . Step 4 For k = 1, 2, ··· until convergence ⎡ ⎤ ⎡ ⎤ vec(X(k)) vec(X(k-1)) ⎢ ⎥ ⎢ ⎥ vec(Y(k)) vec(Y(k-1)) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ − vec(X(k))⎦ ⎣ − vec(X(k-1))⎦ − vec(Y(k)) − vec(Y(k-1)) ⎛ ⎡ ⎤ ⎞ ⎡ ⎤ vec(X(k-1)) vec(F ) ⎜ ⎢ ⎥ ⎟ ⎢ ⎥ vec(Y(k-1)) vec(F ) S S ⎜ 2 1 2 ⎢ ⎥ ⎟ T −1 T ⎢ ⎥ + μ(S S) S − , ⎜ ⎢ ⎥ ⎟ ⎣ ⎦ ⎝ − vec(F ) S S ⎣ − vec(X(k-1))⎦ ⎠ 1 2 1 − vec(F ) − vec(Y(k-1)) 0 <μ< 2 Step 5 If ||X(k) − X(k-1) ||/||X(k)|| <ε, ||X(k) − X(k-1) ||/||X(k)|| <ε, ||Y(k) − Y(k-1) ||/||Y(k)|| <ε and ||Y(k) − Y(k-1) ||/||Y(k)|| <ε stop; otherwise go to step 6. Step 6 Set k = k + 1 and return to step 4. Step 7 End. 4. 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), AX + YB = C, DX + YE = F. Given ⎡ ⎤ ⎡ ⎤ 125 312 13 22 ⎣ ⎦ ⎣ ⎦ A = , D = , B = 341 , E = 413 , 25 13 201 522 (−36 + 38 r,36 − 38 r)(−27 + 30 r,27 − 30 r)(−50 + 33 r,50 − 33 r) C = , (−40 + 38 r,40 − 38 r)(−30 + 24 r,30 − 24 r)(−42 + 27 r,42 − 27 r) (−63 + 54 r,63 − 54 r)(−29 + 21 r,29 − 21 r)(−46 + 34 r,46 − 34 r) F = . (−48 + 33 r,48 − 33 r)(−21 + 13 r,21 − 13 r)(−36 + 21 r,36 − 21 r) 484 A. M. E. BAYOUMI AND M. A. RAMADAN Table 1. The iterative solution for algorithm I for μ = 0.5 kX(k) = (X(k), X(k)) and Y(k) = (Y(k), Y(k)) 1.61 + .238 r - 2.76 + 1.81 r - 4.77 + 1.82 r - .407 - .238 r 2.88 - 1.81 r4.88 - 1.82 r 50 X(k) = X(k) = - 4.28 + 5.20 r - 1.23 + 1.51 r - 2.70 + 2.16 r 4.15 - 5.20 r 1.58 - 1.51 r3.29 - 2.16 r - 3.96 + 2.44 r - 1.66 + 3.82 r - 5.40 + 2.66 r 4.45 - 2.44 r 1.82 - 3.82 r5.66 - 2.66 r Y(k) = , Y(k) = .8978 - .151 r - 2.13 + 1.53 r - 4.78 + 1.41 r - .363 + .151 r 2.54 - 1.53 r4.41 - 1.41 r - .516 + 1.68 r - 2.95 + 1.97 r - 4.95 + 1.97 r .737 - 1.68 r 2.97 - 1.97 r4.97 - 1.97 r 100 X(k) = , X(k) = - 4.05 + 5.03 r - 1.86 + 1.91 r - 3.76 + 2.84 r 4.02 - 5.03 r1.92 - 1.91 r3.87 - 2.84 r - 4.80 + 2.89 r - 1.93 + 3.96 r - 5.90 + 2.95 r 4.89 - 2.89 r 1.96 - 3.96 r5.95 - 2.95 r Y(k) = , Y(k) = - .646 + .810 r - 2.84 + 1.91 r - 4.14 + 1.07 r .811 - .810 r 2.91 - 1.91 r4.07 - 1.07 r - .911 + 1.94 r - 2.99 + 1.99 r - 4.99 + 1.99 r .951 - 1.94 r 2.99 - 1.99 r4.99 - 1.99 r 150 X(k) = , X(k) = - 4.010 + 5.00 r - 1.97 + 1.98 r - 3.95 + 2.97 r 4.00 - 5.00 r1.98 - 1.98 r3.97 - 2.97 r - 4.96 + 2.98 r - 1.98 + 3.99 r - 5.98 + 2.99 r 4.98 - 2.98 r 1.99 - 3.99 r5.99 - 2.99 r Y(k) = , Y(k) = - .935 + .965 r - 2.97 + 1.98 r - 4.02 + 1.01 r .965 - .965 r 2.98 - 1.98 r4.01 - 1.01 r - 1.02 + 2.01 r - 3.00 + 2.00 r - 5.00 + 2.00 r 1.01 - 2.01 r 3.00 - 2.00 r5.00 - 2.00 r 195 X(k) = , X(k) = - 3.99 + 4.99 r - 2.00 + 2.00 r - 4.01 + 3.00 r 3.99 - 4.99 r 2.00 - 2.00 r4.00 - 3.00 r - 5.00 + 3.00 r - 2.00 + 4.00 r - 6.00 + 3.00 r 5.00 - 3.00 r 2.00 - 4.00 r6.00 - 3.00 r Y(k) = , Y(k) = - 1.01 + 1.00 r - 3.00 + 2.00 r - 3.99 + .99 r 1.00 - 1.00 r 3.00 - 2.00 r 3.99 - .99 r This system of coupled fuzzy Sylvester matrix equations has a unique solution (−1 + 2 r,1 − 2 r)(−3 + 2 r,3 − 2 r)(−5 + 2 r,5 − 2 r) X = , (−4 + 5 r,4 − 5 r)(−2 + 2 r,2 − 2 r)(−4 + 3 r,4 − 3 r) (−5 + 3 r,5 − 3 r)(−2 + 4 r,2 − 4 r)(−6 + 3 r,6 − 3 r) Y = . (−1 + r,1 − r)(−3 + 2 r,3 − 2 r)(−4 + r,4 − r) Algorithm I and algorithm II are applied to solve generalized Sylvester matrix equations (1). (1, 1)(1, 1)(1, 1) When selecting the initial matrices as X(0), Y(0) = . Algorithm I is (1, 1)(1, 1)(1, 1) 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 solu- tion 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 giveninTable 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), AX + YB = C, DX + YE = F. FUZZY INFORMATION AND ENGINEERING 485 Table 2. The iterative solution for algorithm II for μ = 0.5. kX(k) = (X(k), X(k)) and Y(k) = (Y(k), Y(k)) - 1.71 + 2.38 r - 3.07 + 2.03 r - 5.06 + 2.03 r 1.38 - 2.38 r 3.03 - 2.03 r5.03 - 2.03 r 50 X(k) = X(k) = - 3.919 + 4.95 r - 2.20 + 2.10 r - 4.34 + 3.18 r 3.95 - 4.95 r2.10 - 2.10 r4.18 - 3.18 r - 4.805 + 2.89 r - 1.93 + 3.96 r - 5.90 + 2.95 r 4.89 - 2.89 r 1.96 - 3.96 r5.95 - 2.95 r Y(k) = , Y(k) = - .646 + .810 r - 2.84 + 1.91 r - 4.14 + 1.07 r .8117 - .810 r2.91 - 1.91 r4.07 - 1.07 r - 1.02 + 2.01 r - 3.00 + 2.00 r - 5.00 + 2.00 r 1.01 - 2.01 r 3.00 - 2.00 r5.00 - 2.00 r 100 X(k) = , X(k) = - 3.99 + 4.99 r - 2.00 + 2.00 r - 4.01 + 3.00 r 3.99 - 4.99 r2.00 - 2.00 r4.00 - 3.00 r - 4.99 + 2.99 r - 1.99 + 3.99 r - 5.99 + 2.99 r 4.99 - 2.99 r1.99 - 3.99 r5.99 - 2.99 r Y(k) = , Y(k) = - .988 + .993 r - 2.99 + 1.99 r - 4.00 + 1.00 r .993 - .993 r2.99 - 1.99 r4.00 - 1.00 r - 1.00 + 2.00 r - 3.00 + 2.00 r - 5.00 + 2.00 r 1.00 - 2.00 r 3.00 - 2.00 r5.00 - 2.00 r 120 X(k) = , X(k) = - 3.99 + 5.00 r - 2.00 + 2.00 r - 4.00 + 3.00 r 4.00 - 5.00 r 2.00 - 2.00 r4.00 - 3.00 r - 4.99 + 2.99 r - 1.99 + 4.00 r - 5.99 + 3.00 r 4.99 - 2.99 r 2.00 - 4.00 r6.00 - 3.00 r Y(k) = , Y(k) = - .99 + .99 r - 2.99 + 1.99 r - 4.00 + 1.00r .99 - .99 r 2.99 - 1.99 r 4.00 - 1.00r Given 2 −1 2 −3 2 −3 3 −1 A = , D = , B = , E = , −33 3 −2 −13 −25 (7 r,18 − 7 r)(−8 + 13 r,13 − 13 r) C = , (−17 + 10 r,15 − 10 r)(−27 + 18 r,6 − 18 r) (−8 + 12 r,23 − 12 r)(−4 + 19 r,27 − 19 r) F = . (−3 + 12 r,32 − 12 r)(−18 + 18 r,23 − 18 r) This system of coupled fuzzy Sylvester matrix equations has a unique solution (1 + r,5 − r)(2 + r,4 − r) (3 + r,4 − r)(1 + 2 r,5 − 2 r) X = , Y = (−1 + r,3 − r)(1 + 2 r,3 − 2 r) (2 + r,3 − r)(−3 + 2 r,3 − 2 r) We use algorithm III to solve generalized Sylvester matrix equations (1). ⎡ ⎤ ⎡ ⎤ 20002000 01000010 ⎢ ⎥ ⎢ ⎥ 03000200 30000001 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 00200030 00013000 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00030003⎥ ⎢ 00300300⎥ S = ⎢ ⎥ , S = ⎢ ⎥ , 1 2 ⎢ 20003000⎥ ⎢ 03000020⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 30000300 02000002 ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ 00200050 00031000 00300005 00020100 † † † † S S T + Q T − Q 1 2 T = S + S , Q = S − S , S = , S = 1 2 1 2 † † † † S S 2 T − Q T + Q 2 1 486 A. M. E. BAYOUMI AND M. A. RAMADAN ⎡ ⎤ ⎡ ⎤ 7r 1 + r ⎢ ⎥ ⎢ ⎥ −1 + r −17 + 10 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 + r −8 + 13 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 + 2 r −27 + 18 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 3 + r ⎢ −8 + 12 r ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −3 + 12 r ⎥ ⎢ 2 + r ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ vec(C) −4 + 19 r 1 + 2 r vec(X) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ vec(Y) vec(F) −18 + 18 r −3 + 2 r † † † ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ M = = S N = S = S = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ − vec(X) − vec(C) −18 + 7 r −5 + r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − vec(Y) − vec(F) −15 + 10 r −3 + r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −13 + 13 r −4 + r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −6 + 18 r ⎥ ⎢ −3 + 2 r⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −23 + 12 r −4 + r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −32 + 12 r −3 + r ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ −27 + 19 r −5 + 2 r −23 + 18 r −3 + 2 r (0, 0)(0, 0) When the initial matrices are chosen as X(0), Y(0) = . Algorithm III is (0, 0)(0, 0) convergent for 0 <μ< 0.5. After 118 iterations we obtain (1.0000 + 1.0000 r, 5.0000 − 1.0000 r)(2.0000 + 1.0000 r, 3.9999 − 0.9999 r) X = , (−1.0000 + 1.0000 r, 3.0000 − 0.9999 r)(0.9999 + 2.0000 r, 2.9999 − 1.9999 r) (2.9999 + 0.9999 r, 3.9999 − 0.9999 r)(0.9999 + 2.0000 r, 5.0000 − 1.9999 r) Y = (1.9999 + 0.9999 r, 3.0000 − 0.9999 r)(−3.0000 + 2.0000 r, 3.0000 − 2.0000 r) Example 4.3: In this example, we illustrate our theoretical results of algorithm V for solving coupled fuzzy Sylvester matrix equations given in (1), AXB + CYD = F , EXG + HYN = F . 1 2 Given 2 −2 −1 −3 21 −23 A = , C = , E = , H = , −34 2 −4 −12 1 −2 32 −12 3 −1 B = , D = , G = , 2 −1 2 −2 0 −1 5 −3 (−47 + 42 r,37 − 33 r)(−58 + 34 r,14 − 30 r) N = , F = , 10 (−37 + 55 r,99 − 71 r)(−51 + 48 r,62 − 57 r) (−17 + 49 r,68 − 42 r)(−40 + 22 r,5 − 29 r) F = (−26 + 27 r,37 − 38 r)(−17 + 21 r,16 − 15 r) This system of coupled fuzzy Sylvester matrix equations has a unique solution (1 + r,3 − r)(1 + 2 r,4 − r) X = , (2 + r,5 − 2 r)(1 + r,3 − r) FUZZY INFORMATION AND ENGINEERING 487 (3 + r,5 − 2 r)(2 + r,3 − r) Y = . (2 + r,4 − r)(−1 + r,3 − r) We apply algorithm V to solve the generalized Sylvester matrix equations (2). ⎡ ⎤ ⎡ ⎤ 604 0 130 0 0604 0 0 26 ⎢ ⎥ ⎢ ⎥ 012080 4 4 0 9060 2 0 08 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 400 2 002 6 0420 2 6 00 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 083 0 400 8 6004 0 8 40 ⎢ ⎥ ⎢ ⎥ S = ⎢ ⎥ , S = ⎢ ⎥ , 1 2 ⎢ 6 3 0 001503⎥ ⎢ 000010 0 20⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 060 0 501 0 3000 0 1002 ⎢ ⎥ ⎢ ⎥ ⎦ ⎦ ⎣ ⎣ 000 0 600 0 2121 0 9 00 101 0 060 0 0202 3 0 00 † † † † S S 1 T + Q T − Q 1 2 T = S + S , Q = S − S , S = , S = 1 2 1 2 † † † † S S T − Q T + Q 2 1 ⎡ ⎤ ⎡ ⎤ −47 + 42 r 0.9999 + 1.0000 r ⎢ ⎥ ⎢ ⎥ −37 + 55 r 2.0000 + 0.9999 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −58 + 34 r 0.9999 + 2.0000 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1.0000 + 0.9999 r −51 + 48 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 3.0000 + 0.9999 r ⎢ −17 + 49 r⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1.9999 + 1.0000 r ⎥ ⎡ ⎤ −26 + 27 r ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ vec(F ) ⎢ ⎥ ⎢ ⎥ 1.9999 + 1.0000 r vec(X) −40 + 22 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ vec(F ) −0.9999 + 0.9999 r vec(Y) 2 −17 + 21 r † † † ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ M = = S N = S = S = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ −37 + 33 r −3.000 + 1.0000 r − vec(X) − vec(F ) 1 ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ −99 + 71 r −4.9999 + 1.9999 r − vec(Y) ⎢ ⎥ ⎢ ⎥ − vec(F ) ⎢ ⎥ ⎢ ⎥ −14 + 30 r −4.0000 + 1.0000 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −62 + 57 r⎥ ⎢ −2.9999 + 0.9999 r⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −68 + 42 r −4.9999 + 1.9999 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −37 + 38 r −4.0000 + 1.0000 r ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ −5 + 29 r −3.000 + 1.0000 r −16 + 15 r −2.9999 + 0.9999 r (0, 0)(0, 0) When the initial matrices are chosen as X(0), Y(0) = .ThealgorithmVis (0, 0)(0, 0) convergent for 0 <μ< 0.7. After 145 iterations we obtain (1.0000 + 0.9999 r, 3.0000 − 1.0000 r)(0.9999 + 2.0000 r, 4.0000 − 1.0000 r) X = , (2.0000 + 0.9999 r, 4.9999 − 1.9999 r)(1.0000 + 0.9999 r, 2.9999 − 0.9999 r) (3.0000 + 0.9999 r, 4.9999 − 1.9999 r)(1.9999 + 1.0000 r, 3.0000 − 1.0000 r) Y = , (1.9999 + 1.0000 r, 4.0000 − 1.0000 r)(−0.9999 + 0.9999 r, 2.9999 − 0.9999 r) 5. 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 488 A. M. E. BAYOUMI AND M. A. RAMADAN 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 cou- pled 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 itera- tive 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. Acknowledgments The authors would like to express their heartfelt thanks to the editor and anonymous referees for their useful comments. Conflict of interest The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding This research received no specific grant from any funding agency in the public, commercial, or not- for-profit sectors. ORCID Ahmed M. E. Bayoumi http://orcid.org/0000-0002-7691-4111 References [1] Ding F, Liu PX, Ding J. Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle. Appl Math Comput. 2008;197(1):41–50. [2] Ding F, Chen T. Iterative least squares solutions of coupled Sylvester matrix equations. Syst Cont Lett. 2005;54(2):95–107. [3] Ding F, Chen T. On iterative solutions of general coupled matrix equations. SIAM J Cont Optim. 2006;44(6):2269–2284. [4] Ding F, Chen T. Gradient based iterative algorithms for solving a class of matrix equations. IEEE Trans Automat Cont. 2005;50(8):1216–1221. [5] Hajarian M. Generalized reflexive and anti-reflexive solutions of the coupled Sylvester matrix equations via CD algorithm. J Vib Cont. 2018;24(2):343–356. [6] Zhang H. Iterative solutions of a set of matrix equations by using the hierarchical identification principle. Abst Appl Analy. 2014: 1–10. doi:10.1155/2014/649524. Article ID 649524. [7] Ramadan MA, El-Danaf TS, E AM. Bayoumi, a modified gradient-based algorithm for solving extended Sylvester-conjugate matrix equations. Asian J Cont. 2018;20(1): 228–235. [8] Friedman M, Ming M, Kandel A. Fuzzy linear systems. Fuzzy Set Syst. 1998;96:201–209. [9] Ma M, Friedman M, Kandel A. A new fuzzy arithmetic. Fuzzy Set Syst. 1999;108:83–90. [10] Sadeghi A, Abbasbandy S, Abbasnejad ME. The common solution of the pair of fuzzy matrix equations. World Appl Sci J. 2011;15(2):232–238. FUZZY INFORMATION AND ENGINEERING 489 [11] Bayoumi AME. Finite iterative Hamiltonian solutions of the generalized coupled Sylvester - conjugate matrix equations. Trans Ins Meas Cont. 2019;41(4):1139–1148. [12] Bayoumi AME, Ramadan MA. Finite iterative Hermitian R-conjugate solutions of the generalized coupled Sylvester- conjugate matrix equations. Comp Math Appl. 2018;75:3367–3378. [13] Behera D, Chakraverty S. Solution to Fuzzy System of Linear Equations with Crisp Coefficients. Fuzzy Inform Eng. 2013;5(2):205–219. [14] Wang QW, Lv RY, Zhang Y. The least-squares solution with the least norm to a system of tensor equations over the quaternion algebra. Lin Multilin Algebra. 2020. doi:10.1080/03081087.2020. [15] Ramadan MA, El-Danaf TS, Bayoumi AME. Two iterative algorithms for the reflexive and Hermetian reflexive solutions of the generalized Sylvester matrix equation. J Vib Cont. 2015;21(3):483–492. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

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

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FUZZY INFORMATION AND ENGINEERING 2020, VOL. 12, NO. 4, 464–489 https://doi.org/10.1080/16168658.2021.1923442 Solving Two Coupled Fuzzy Sylvester Matrix Equations Using Iterative Least-squares Solutions a b Ahmed M. E. Bayoumi and Mohamed A. Ramadan a b Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt; Department of Mathematics, Faculty of Science, Menoufia University, Shebeen El- Koom, Egypt ABSTRACT ARTICLE HISTORY Received 21 January 2019 In this paper, five iterative methods for solving two coupled fuzzy Revised 10 July 2020 Sylvester matrix equations are considered. The two coupled fuzzy Accepted 26 April 2021 Sylvester matrix equations are expressed by using the generalized inverse of the coefficient matrix, then iterative solutions are con- KEYWORDS structed by applying the hierarchical identification principle and by Coupled fuzzy Sylvester using the block-matrix inner product (the star product for short). A matrix equations; Iterative proposed modification to this algorithm to solve the first coupled algorithm; Kronecker product; Frobenius norm; fuzzy Sylvester matrix equations is suggested. This proposed modifi- Star product cation 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. 1. 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 AX B = F and the generalized Sylvester matrix equation AX B + CX D = F, by extending Jacobi and Gauss–Seidel iteration methods for Ax = b. Ding and Chen [2] suggested a least-squares iterative algorithm to solve the generalized coupled Sylvester matrix equation AX + YB = C, DX + YE = F (1) 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 algo- rithms [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 gen- eralized reflexive solution X and the generalized anti-reflexive solution Y of the coupled CONTACT Ahmed M. E. Bayoumi ame_bayoumi@yahoo.com © 2021 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society of China & Operations Research Society of Guangdong Province. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 465 Sylvester matrix equations AXB + CYD = F , EXG + HYN = F.(2) 1 2 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 col- umn. 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 lin- ear 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, defi- nitions, 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 equa- tions 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 sec- ond 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 prop- erties of these iterative algorithms. In Section4, numerical examples are introduced to illustrate the effectiveness of the proposed algorithms. 2. Preliminaries The following notations, definitions, lemmas and theorems will be used to develop the proposed work. We use A to denote the transpose of A.Thesetofall m × n real matri- m×n m×n T T T T ces is denoted by R .For A ∈ R , vec (A) is defined as vec (A) = [a a ······ a ] 1 2 n where a is the ith column of the matrix A. The Kronecker product of two matrices A = (a ) and B is denoted by A ⊗ B. We have the following well-known property ij m×n vec (MX N) = (N ⊗ M) vec (X) for matrices M, X, N. 466 A. M. E. BAYOUMI AND M. A. RAMADAN Definition 2.1: Block-matrix inner product [2] The block-matrix inner product is called the star product for short, denoted by (∗).Let ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ X Y A A ··· A 1 1 11 12 1p ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ X Y A A ··· A 2 2 21 22 2p ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ mp×n np×m X = ⎢ ⎥ ∈ R , Y = ⎢ ⎥ ∈ R , S = ⎢ ⎥ , . . A . . . . . . . . . ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ . . . . . X Y A A ··· A p p p1 p2 pp ⎡ ⎤ B B ··· B 11 12 1p ⎢ ⎥ B B ··· B 21 22 2p ⎢ ⎥ S = ⎢ ⎥ . B . . . . . . . ⎣ ⎦ . . . B B ··· B p1 p2 pp Then the block-matrix star product is defined as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ X Y Y A X A X ··· A X 1 1 1 11 1 12 2 1p p ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ X Y Y A X A X ··· A X 2 2 2 21 1 22 2 2p p ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ X ∗ Y = ∗ = , S ∗ X = , ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ . . . A . . . . . . . . . . ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ . . . . . . X Y Y A X A X ··· A X p p p p1 1 p2 2 pp p ⎡ ⎤ ⎡ ⎤ X B X B ··· X B A B A B ··· A B 1 11 1 12 1 1p 11 11 12 12 1p 1p ⎢ ⎥ ⎢ ⎥ X B X B ··· X B A B A B ··· A B 2 21 2 22 2 2p 21 21 22 22 2p 2p ⎢ ⎥ ⎢ ⎥ X ∗ S = , S ∗ S = . ⎢ ⎥ ⎢ ⎥ B . . . A B . . . . . . . . . . . . . ⎣ ⎦ ⎣ ⎦ . . . . . . . . X B X B ··· X B A B A B ··· A B p p1 p p2 p pp p1 p1 p2 p2 pp pp The following basic concepts of fuzzy number arithmetic and fuzzy linear system of equa- tions 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: 1) u(r) is a bounded left continuous non-decreasing function over [0, 1], 2) u(r) is a bounded right continuous non-increasing function over [0, 1], 3) u(r) ≤ u(r),0 ≤ r ≤ 1. A crisp number α is simply represented by u(r) = u(r) = α,0 ≤ r ≤ 1. The triangular fuzzy numbers are very popular and denoted by u = (c, α, β) and defined by x − c + α c − α ≤ x ≤ c, ⎨ α c + β − x u(x) = c ≤ x ≤ c + β, 0 otherwise. where α> 0and β> 0. The parametric form of the number is u(r) = rα + c − α, u(r) = c + β − β r. FUZZY INFORMATION AND ENGINEERING 467 The addition and scalar multiplication of fuzzy numbers are defined by the extension principle and can be equivalently represented as follows, see [8, 9]. For arbitrary fuzzy numbers v = (v (r), v (r)) and w = (w (r), w (r)),0 ≤ r ≤ 1and real number k as follows: a) v = w if and only if v (r) = w (r) and v (r) = w (r), b) v + w = ( v (r) + w (r), v (r) + w (r)), c) v − w = ( v (r) − w (r), v (r) − w (r)), (kv (r), k v (r)) k ≥ 0, d) kv = ( k v (r) , kv (r)) k < 0. Definition 2.3: Consider the p × q linear system of equations a v + a v + ··· + a v = w , ⎪ 11 1 12 2 1q q 1 a v + a v + ··· + a v = w , 21 1 22 2 2n q 2 (3) ⎪ . a v + a v + ··· + a v = w , p1 1 p2 2 pq q p p×q T where the coefficient matrix A = (a ) ∈R is given crisp matrix and w = (w , w , ... , w ) ij 1 2 p is given vector of fuzzy numbers and v = (v , v , ... , v ) is vector of fuzzy numbers to be 1 2 q determined. This system is called an FSLE. Definition 2.4: A fuzzy number vector v = (v , v , ... , v ) where v = (v (r), v (r)),0 ≤ 1 2 q i i r ≤ 1, i = 1, 2, ... , q, is called a solution of the fuzzy linear system of equations (3) if q q ⎪ a v = a v = w , ij j ij j i j=1 j=1 (4) ⎪ q q ⎪ a v = a v = w ij j ij j i j=1 j=1 In general, an arbitrary equation for either w or w is a linear combination of v ’s and v ’s, i i j j respectively. Therefore, in order to solve Equation (3) one must solve a 2p × 2q crisp linear system of equations (5) as follows: Sv = w (5) where S S v w 1 2 S = , v = , w =.(6) S S v w 2 1 where the element of S = (s ),1 ≤ i, j ≤ 2q, as follows: ij if a ≥ 0 ⇒ s = a , s = a ij i,j i,j i+p,j+q i,j (7) if a < 0 ⇒ s =−a , s =−a ij i,j+q ij i+p,j i,j 468 A. M. E. BAYOUMI AND M. A. RAMADAN Theorem 2.1: [10]: Let matrix S be in the form (6), then the matrix {1,3} {1,3} {1,3} {1,3} 1 (S + S ) + (S − S ) (S + S ) − (S − S ) 1 2 1 2 1 2 1 2 {1,3} S = (8) {1,3} {1,3} {1,3} {1,3} (S + S ) − (S − S ) (S + S ) + (S − S ) 1 2 1 2 1 2 1 2 {1,3} {1,3} is a {1, 3}-inverse of the matrix S, where (S + S ) and (S − S ) are {1, 3}-inverse of 1 2 1 2 the matrices (S + S ) and (S − S ), respectively. In particular, the Moore–Penrose inverse 1 2 1 2 of the matrix S is † † † † (S + S ) + (S − S ) (S + S ) − (S − S ) 1 2 1 2 1 2 1 2 S = .(9) † † † † 2 (S + S ) − (S − S ) (S + S ) + (S − S ) 1 2 1 2 1 2 1 2 Theorem 2.2: [10]: {1,3} For the consistent system (5) and any {1, 3}-inverse S of the coefficient matrix S, {1,3} v = S 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 x(k) = x k→∞ T −1 T x(k) = x(k − 1) + μ(A A) A [b − Ax(k − 1)], 0 <μ< 2. 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): T −1 T T T −1 X(k) = X(k − 1) + μ(A A) A [F − AX(k − 1)B]B (BB ) ,0 <μ< 2. Lemma 2.3: [2] The coupled fuzzy Sylvester matrix equations given in (1), AX + YB = C, DX + YE = F, m×m n×n m×n where A, D ∈R and B, E ∈R are given crisp matrices and C, F ∈R are given fuzzy m×n matrices while X, Y ∈R are fuzzy matrices to be determined. Equation (1) has a unique solution if and only if the matrix I ⊗AB ⊗ I n m 2mn×2mn Q = ∈ R I ⊗DE ⊗ I n m is non-singular; in this case, the unique solution is given by vec(X) vec(C) −1 = Q vec(Y) vec(F) and the corresponding homogeneous matrix equation AX + YB = 0, DX + YE = 0has a unique solution X = Y = 0. FUZZY INFORMATION AND ENGINEERING 469 Lemma 2.4: The coupled Sylvester matrix equations given in (2), AXB + CYD = F , EXG + HYN = F . 1 2 m×m n×l m×l where A, C, E, H ∈R and B, D, G, N ∈R are given crisp matrices and F , F ∈R are 1 2 m×n given fuzzy matrices while X, Y ∈R are fuzzy matrices to be determined. Equation (2) has a unique solution if and only if the matrix T T B ⊗AD ⊗ C 2lm×2nm Q = ∈ R T T G ⊗EN ⊗ H is non-singular; in this case, the unique solution is given by vec(X) vec(F ) −1 = Q vec(Y) vec(F ) and the corresponding homogeneous matrix equation AXB + CYD = 0, EXG + HYN = 0 has a unique solution X = Y = 0. 3. 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]. 3.1. 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), AX + YB = C, DX + YE = F m×m n×n m×n where A, D ∈R and B, E ∈R are given crisp matrices and C, F ∈ R are given fuzzy m×n matrices while X, Y ∈ R are fuzzy matrices to be determined. The basic idea is to regard Equation (1) as two matrices C − YB R = (10) F − YE R = C − AX, F − DX (11) Hence, Equation (1) can be decomposed into two matrix equations of the form: K X = R (12) 1 1 YK = R (13) 2 2 Here K = and K = B, E 1 2 Then, we can define the following iterative formulas T −1 T X(k) = X(k − 1) + μ(K K ) K (R − K X(k − 1)) (14) 1 1 1 1 1 470 A. M. E. BAYOUMI AND M. A. RAMADAN T T −1 Y(k) = Y(k − 1) + μ(R − Y(k − 1) K ) K (K K ) (15) 2 2 2 2 2 where μ is the convergence factor. Substituting from Equations (10) and (11) into Equa- tions (14) and (15) gives A C − AX(k − 1) − YB T −1 X(k) = X(k − 1) + μ(K K ) , (16) D F − DX(k − 1) − YE T −1 Y(k) = Y(k − 1) + μ C − AX − Y(k − 1)B, F − DX − Y(k − 1)E B, E (K K ) (17) 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 hierar- chical 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 A C − AX(k − 1) − Y(k − 1)B T −1 X(k) = X(k − 1) + μ(K K ) , (18) D F − DX(k − 1) − Y(k − 1)E Y(k) = Y(k − 1) + μ C − AX(k − 1) − Y(k − 1)B, F − DX(k − 1) − Y(k − 1)E T −1 × B, E (K K ) (19) 2 2 μ = or μ = (20) −1 −1 m + n T T T T λ (K (K K ) K ) + λ (K (K K ) K ) max 1 1 max 2 2 1 1 2 2 In this case, the iterative least-squares solutions of coupled fuzzy Sylvester matrix equa- tions can be written as A C − AX(k − 1) − Y(k − 1)B T −1 X(k) = X(k − 1) + μ(K K ) , (21) D F − DX(k − 1) − Y(k − 1)E A C − AX(k − 1) − Y(k − 1)B T −1 X(k) = X(k − 1) + μ(K K ) , (22) D F − DX(k − 1) − Y(k − 1)E Y(k) = Y(k − 1) + μ C − AX(k − 1) − Y(k − 1)B, F − DX(k − 1) − Y(k − 1)E T −1 × B, E (K K ) (23) Y(k) = Y(k − 1) + μ C − AX(k − 1) − Y(k − 1)B, F − DX(k − 1) − Y(k − 1)E T −1 × B, E (K K ) (24) 2 2 μ = or μ = −1 −1 T T T T m + n λ (K (K K ) K ) + λ (K (K K ) K ) max 1 1 max 2 2 1 1 2 2 We now outline our suggested algorithm. Algorithm I m×m n×n m×n Step 1 Input crisp matrices A, D ∈R and B, E ∈R and input fuzzy matrices C, F ∈R , and number ε. FUZZY INFORMATION AND ENGINEERING 471 m×n Step 2 Given any two initial fuzzy matrices X(0), Y(0) ∈R . Step 3 Compute K = and K = B, E 1 2 Step 4 For k = 1, 2, ··· until convergence A C − AX(k − 1) − Y(k − 1)B T −1 X(k) = X(k − 1) + μ(K K ) , D F − DX(k − 1) − Y(k − 1)E A C − AX(k − 1) − Y(k − 1)B T −1 X(k) = X(k − 1) + μ(K K ) , D F − DX(k − 1) − Y(k − 1)E Y(k) = Y(k − 1) + μ C − AX(k − 1) − Y(k − 1)B, F − DX(k − 1) − Y(k − 1)E T −1 × B, E (K K ) Y(k) = Y(k − 1) + μ C − AX(k − 1) − Y(k − 1)B, F − DX(k − 1) − Y(k − 1)E T −1 × (K K ) B, E 2 2 μ = or μ = −1 −1 T T T T m + n λ (K (K K ) K ) + λ (K (K K ) K ) max 1 1 max 2 2 1 1 2 2 Step 5 If ||X(k) − X(k-1) ||/||X(k)|| <ε, ||X(k) − X(k-1) ||/||X(k)|| <ε, ||Y(k) − Y(k-1) ||/||Y(k)|| <ε and ||Y(k) − Y(k-1) ||/||Y(k)|| <ε stop; otherwise go to step 6. Step 6 Set k = k + 1 and return to step 4. Step 7 End. Theorem 3.1: If the coupled fuzzy Sylvester matrix equations (1) is consistent and has a ∗ ∗ m×n ∗ ∗ m×n ∗ ∗ unique fuzzy solutions X = (X , X ) ∈R and Y = (Y , Y ) ∈R and 2 2 μ = or μ = −1 −1 T T T T m + n λ (K (K K ) K ) + λ (K (K K ) K ) max 1 1 max 2 2 1 1 2 2 then the iterative sequence {X(k)}, {X(k)}, {Y(k)} and {Y(k)} generated by algorithm I con- ∗ ∗ ∗ ∗ ∗ ∗ ∗ verges to X , X , Y and Y , that is, lim X(k) = X ,lim X(k) = X ,lim Y(k) = Y and k→∞ k→∞ k→∞ lim Y(k) = Y for any initial fuzzy matrices X(0) , X(0) , Y(0) and Y(0) . k→∞ Proof: First, we define the estimation error matrices as ∗ ∗ ∗ ∗ ξ (k) = X(k) − X , ξ (k) = Y(k) − Y , ξ (k) = X(k) − X and ξ (k) = Y(k) − Y for 1 2 3 4 k = 1, 2, ··· . Using algorithm I and the above error matrices, we can obtain A Aξ (k − 1) + ξ (k − 1)B 1 2 T −1 ξ (k) = ξ (k − 1) − μ(K K ) (25) 1 1 1 D Dξ (k − 1) + ξ (k − 1)E 1 2 ξ (k) = ξ (k − 1) − μ Aξ (k − 1) + ξ (k − 1)B, Dξ (k − 1) + ξ (k − 1)E 2 2 1 2 1 2 T −1 × (K K ) (26) B, E 2 472 A. M. E. BAYOUMI AND M. A. RAMADAN Now, by taking the norm of (25) and (26) and using the following formula, we have −1 −1 −1 T 2 T T T ||K [X + (K K ) Y ] || = tr{[X + (K K ) Y] (K K )[X + (K K ) Y] } 1 1 1 1 1 1 1 1 1 −1 T T T T T = tr {X (K K ) X + 2 X Y + Y (K K ) Y } 1 1 1 1 −1 2 T T =||K X|| + 2 tr[ X Y ] + (K K ) Y . (27) 1 1 Gives 2 T T ||K ξ (k)|| = tr{ξ (k) K K ξ (k)} 1 1 1 1 1 1 T T = tr{ξ (k − 1) K K ξ (k − 1)} 1 1 1 1 A Aξ (k − 1) + ξ (k − 1)B 1 2 − 2 μ tr ξ (k − 1) D Dξ (k − 1) + ξ (k − 1)E 1 2 −1 Aξ (k − 1) + ξ (k − 1)B 1 2 2 T T + μ (K K ) K 1 1 Dξ (k − 1) + ξ (k − 1)E 1 2 2 T ≤||K ξ (k − 1)|| − 2 μ tr{[Aξ (k − 1)] (Aξ (k − 1) + ξ (k − 1)B) 1 1 1 1 2 + [Dξ (k − 1)] (Dξ (k − 1) + ξ (k − 1)E)} 1 1 2 2 2 2 + μ m [||Aξ (k − 1) + ξ (k − 1)B|| +||Dξ (k − 1) + ξ (k − 1)E|| ] (28) 1 2 1 2 Similarly, 2 T T ||ξ (k) K || = tr{ξ (k) K K ξ (k)} 2 2 2 2 2 2 T T = tr{ξ (k − 1) K K ξ (k − 1)} 2 2 2 2 − 2 μ tr Aξ (k − 1) + ξ (k − 1)B, Dξ (k − 1) + ξ (k − 1)E 1 2 1 2 × B, E ξ (k − 1) + μ Aξ (k − 1) + ξ (k − 1)B, Dξ (k − 1) + ξ (k − 1)E 1 2 1 2 T −1 × B, E (K K ) 2 T ≤||ξ (k − 1) K || − 2 μ tr{[ξ (k − 1)B] (Aξ (k − 1) 2 2 2 1 + ξ (k − 1)B) + [ξ (k − 1)E] (Dξ (k − 1) + ξ (k − 1)E)} 2 2 1 2 2 2 2 + μ n [||Aξ (k − 1) + ξ (k − 1)B|| +||Dξ (k − 1) + ξ (k − 1)E|| ] (29) 1 2 1 2 Define the nonnegative definite function η(k) by 2 2 η(k) =||K ξ (k)|| +||ξ (k) K || . 1 1 2 2 FUZZY INFORMATION AND ENGINEERING 473 From (28) and (29), this function can be computed as 2 T η(k) ≤||K ξ (k − 1)|| − 2 μ tr{[Aξ (k − 1)] (Aξ (k − 1) + ξ (k − 1)B) 1 1 1 1 2 + [Dξ (k − 1)] ( Dξ (k − 1) + ξ (k − 1)E)} 1 1 2 2 2 2 + μ m [||Aξ (k − 1) + ξ (k − 1)B|| +||Dξ (k − 1) + ξ (k − 1)E|| ] 1 2 1 2 2 T +||ξ (k − 1) K || − 2 μ tr{[ξ (k − 1)B] (Aξ (k − 1) + ξ (k − 1)B) 2 2 2 1 2 + [ξ (k − 1)E] (Dξ (k − 1) + ξ (k − 1)E)} 2 1 2 2 2 + μ n [||Aξ (k − 1) + ξ (k − 1)B|| +||Dξ (k − 1) + ξ (k − 1)E|| ] 1 2 1 2 ≤||K ξ (k − 1)|| 1 1 2 T +||ξ (k − 1) K || − 2 μ tr{(Aξ (k − 1) + ξ (k − 1)B) (Aξ (k − 1) + ξ (k − 1)B) 2 2 1 2 1 2 +(Dξ (k − 1) + ξ (k − 1)E) (Dξ (k − 1) + ξ (k − 1)E)} 1 2 1 2 2 2 2 + μ (m + n) [||Aξ (k − 1) + ξ (k − 1)B|| +||Dξ (k − 1) + ξ (k − 1)E|| ] 1 2 1 2 ≤ η(k − 1) − 2 μ[||Aξ (k − 1) + ξ (k − 1)B|| 1 2 +||Dξ (k − 1) + ξ (k − 1)E|| ] 1 2 2 2 2 + μ (m + n) [||Aξ (k − 1) + ξ (k − 1)B|| +||Dξ (k − 1) + ξ (k − 1)E|| ] 1 2 1 2 ≤ η(k − 1) − μ [2 − μ(m + n)][||Aξ (k − 1) + ξ (k − 1)B || 1 2 +||Dξ (k − 1) + ξ (k − 1)E || ] 1 2 k−1 2 2 η(k) ≤ η(1) − μ [2 − μ(m + n) ] [||Aξ (i) + ξ (i)B || +||Dξ (i) + ξ (i)E || ]. 1 2 1 2 i=1 If the convergence factor μ is chosen to satisfy 0 <μ< . m + n Then 2 2 [ ||Aξ (k) + ξ (k)B|| +||Dξ (k) + ξ (k)E|| ] < ∞ 1 2 1 2 k=1 Since the matrix equation (1) has a unique fuzzy solution it follows that as k →∞ lim Aξ (k) + ξ (k)B = 0 and lim Dξ (k) + ξ (k)E = 0. 1 2 1 2 k→∞ k→∞ According to lemma 2.3, we have lim ξ (k) =0and lim ξ (k) = 0. 1 2 k→∞ k→∞ Or ∗ ∗ lim X(k) = X and lim Y(k) = Y k→∞ k→∞ 474 A. M. E. BAYOUMI AND M. A. RAMADAN Similarly, we can prove that ∗ ∗ lim X(k) = X and lim Y(k) = Y . k→∞ k→∞ 3.2. 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), AX + YB = C, DX + YE = F. m×m n×n m×n where A, D ∈R and B, E ∈R are given crisp matrices and C, F ∈R are given fuzzy m×n matrices while X, Y ∈R are fuzzy matrices to be determined. The proposed algorithm is as follows: m×m n×n Algorithm 1: Step 1 Input crisp matrices A, D ∈R and B, E ∈R and input fuzzy m×n matrices C, F ∈R , and number ε. m×n Step 2 Given any two initial fuzzy matrices X(0), Y(0) ∈R . Step 3 Compute K = and K = B, E 1 2 Step 4 For k = 1, 2, ··· until convergence A C − AX(k − 1) − Y(k − 1)B T −1 X(k) = X(k − 1) + μ(K K ) , D F − DX(k − 1) − Y(k − 1)E Y(k) = Y(k − 1) + μ C − AX(k) − Y(k − 1)B, F − DX(k) − Y(k − 1)E T −1 × B, E (K K ) , A C − AX(k − 1) − Y(k − 1)B T −1 X(k) = X(k − 1) + μ(K K ) , D F − DX(k − 1) − Y(k − 1)E Y(k) = Y(k − 1) + μ C − AX(k) − Y(k − 1)B, F − DX(k) − Y(k − 1)E T −1 × B, E (K K ) , 2 2 μ = or μ = −1 −1 T T T T m + n λ (K (K K ) K ) + λ (K (K K ) K ) max 1 1 max 2 2 1 1 2 2 Step 5 If ||X(k) − X(k-1) ||/||X(k)|| <ε, ||X(k) − X(k-1) ||/||X(k)|| <ε, ||Y(k) − Y(k-1) ||/||Y(k)|| <ε and ||Y(k) − Y(k-1) ||/||Y(k)|| <ε stop; otherwise go to step 6. Step 6 Set k = k + 1 and return to step 4. 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). FUZZY INFORMATION AND ENGINEERING 475 3.3. 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 I ⊗AB ⊗ I vec (X) vec (C) n m I ⊗DE ⊗ I vec (Y) vec (F) n m + + + + − − − − If A , B , D , E contain the positive entries of A, B, D, E, respectively, and A , B , D , E contain the absolute value of negative entries of A, B, D, E, respectively, it is obvious that + − + − + − + − A = A − A , B = B − B , D = D − D , E = E − E . So, according to the properties of Kronecker operators it can be written as + − + − I ⊗ A = I ⊗ (A − A ) = I ⊗ A − I ⊗ A , n n n n T T T + − T + − and B ⊗ I = (B − B ) ⊗ I = B ⊗ I − B ⊗ I . m m m m Similarly + − I ⊗ D = I ⊗ D − I ⊗ D , n n n and T T T + − E ⊗ I = E ⊗ I − E ⊗ I . m m m I ⊗AB ⊗ I n m Q = I ⊗DE ⊗ I n m T T + − + − I ⊗ A − I ⊗ A B ⊗ I − B ⊗ I n n m m Q = T T + − + − I ⊗ D − I ⊗ D E ⊗ I − E ⊗ I n n m m Q = S − S (30) 1 2 where T T + + − − I ⊗ A B ⊗ I I ⊗ A B ⊗ I n m n m S = , S = . 1 2 T T + + − − I ⊗ D E ⊗ I I ⊗ D E ⊗ I n m n m Furthermore, it can be concluded that T T + + − − I ⊗ A B ⊗ I I ⊗ A B ⊗ I n m n m T = + T T + + − − I ⊗ D E ⊗ I I ⊗ D E ⊗ I n m n m T = S + S (31) 1 2 Now, the coupled fuzzy Sylvester matrix equations (1) can be written as SM = N (32) T T + + − − S S I ⊗ A B ⊗ I I ⊗ A B ⊗ I 1 2 n m n m S = , S = , S = 1 2 T T + + − − S S 2 1 I ⊗ D E ⊗ I I ⊗ D E ⊗ I n m n m ⎡ ⎤ ⎡ ⎤ vec(X) vec(C) where , ⎢ ⎥ ⎢ ⎥ vec(Y) vec(F) ⎢ ⎥ ⎢ ⎥ M = , N = ⎣ ⎦ ⎣ ⎦ − vec(X) − vec(C) − vec(Y) − vec(F) 476 A. M. E. BAYOUMI AND M. A. RAMADAN By using Lemma 2.1 for the matrix equation (32), then the following least squares based iterative algorithm leads to lim M(k) = M k→∞ ⎡ ⎤ ⎡ ⎤ vec(X(k)) vec(X(k-1)) ⎢ ⎥ ⎢ ⎥ vec(Y(k)) vec(Y(k-1)) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ − vec(X(k)) − vec(X(k-1)) ⎣ ⎦ ⎣ ⎦ − vec(Y(k)) − vec(Y(k-1)) ⎛ ⎡ ⎤ vec(C) vec(X(k-1)) ⎜ ⎢ ⎥ ⎟ ⎢ ⎥ vec(F) S S vec(Y(k-1)) ⎜ ⎢ ⎥ ⎟ T −1 T 1 2 ⎢ ⎥ + μ(S S) S ⎜ − ⎢ ⎥ ⎟ , ⎣ ⎦ ⎝ − vec(C) S S − vec(X(k-1))⎦ ⎠ 2 1 − vec(F) − vec(Y(k-1)) 0 <μ< 2 + + S S I ⊗ A B ⊗ I 1 2 n m Corollary 3.1: Let matrix S be in the form S = where S = , + + S S 2 1 I ⊗ D E ⊗ I n m − − I ⊗ A B ⊗ I n m S = , then the matrix − − I ⊗ D E ⊗ I n m {1,3} {1,3} {1,3} {1,3} T + Q T − Q {1,3} S = (33) {1,3} {1,3} {1,3} {1,3} T − Q T + Q {1,3} {1,3} is a {1, 3}-inverse of the matrix S, where T and Q are {1, 3}-inverse of the matrices T and Q, respectively. In particular, the Moore–Penrose inverse of the matrix S is: † † † † T + Q T − Q S = (34) † † † † 2 T − Q T + Q We now outline our suggested algorithm. Algorithm III m×m n×n Step 1 Input crisp matrices A, D ∈R and B, E ∈R and input fuzzy matrices m×n C, F ∈R , and number ε. m×n Step 2 Given any two initial fuzzy matrices X(0), Y(0) ∈R . Step 3 Compute T T + + − − I ⊗ A B ⊗ I I ⊗ A B ⊗ I S S n m n m 1 2 S = , S = , S = . 1 2 T T + + − − S S I ⊗ D E ⊗ I I ⊗ D E ⊗ I 2 1 n m n m + + + + where A , B , D , E contain the positive entries of A, B, D, E, respectively, and − − − − A , B , D , E contain the absolute value of negative entries of A, B, D, E, respectively, + − + − + − + − where A = A − A , B = B − B , D = D − D , E = E − E . FUZZY INFORMATION AND ENGINEERING 477 Step 4 For k = 1, 2, ··· until convergence ⎡ ⎤ ⎡ ⎤ vec(X(k)) vec(X(k-1)) ⎢ ⎥ ⎢ ⎥ vec(Y(k)) vec(Y(k-1)) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ − vec(X(k)) − vec(X(k-1)) ⎣ ⎦ ⎣ ⎦ − vec(Y(k)) − vec(Y(k-1)) ⎛ ⎡ ⎤ ⎞ ⎡ ⎤ vec(C) vec(X(k-1)) ⎜ ⎢ ⎥ ⎟ ⎢ ⎥ vec(F) S S vec(Y(k-1)) ⎜ ⎢ ⎥ ⎟ 1 2 T −1 T ⎢ ⎥ + μ(S S) S ⎜ − ⎢ ⎥ ⎟ , ⎣ ⎦ − vec(C) S S − vec(X(k-1)) ⎝ ⎣ ⎦ ⎠ 2 1 − vec(F) − vec(Y(k-1)) 0 <μ< 2 Step 5 If ||X(k) − X(k-1) ||/||X(k)|| <ε, ||X(k) − X(k-1) ||/||X(k)|| <ε, ||Y(k) − Y(k-1) ||/||Y(k)|| <ε and ||Y(k) − Y(k-1) ||/||Y(k)|| <ε stop; otherwise go to step 6. Step 6 Set k = k + 1 and return to step 4. Step 7 End. 3.4. 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), AXB + CYD = F , EXG + HYN = F . 1 2 m×m n×l m×l where A, C, E, H ∈R and B, D, G, N ∈ R are given crisp matrices and F , F ∈ R are 1 2 m×n given fuzzy matrices while X, Y ∈ R are fuzzy matrices to be determined. The basic idea is to regard Equation (2) as two matrices F − CYD R = (35) F − HYN R = F − AXB, F − EXG (36) 2 1 2 Hence, Equation (2) can be decomposed into two matrix equations of the form: S X ∗ T1 = R (37) 1 1 S ∗ YT = R (38) 2 2 2 A B where S = and T = 1 1 E G where S = C, H and T = D, N 2 2 Then we can define the following iterative formulas T −1 T T T −1 X(k) = X(k − 1) + μ(S S ) S [R − S X(k − 1) ∗ T1] ∗ T ( T T ) (39) 1 1 1 1 1 1 1 1 T −1 T T T −1 Y(k) = Y(k − 1) + μ(S S ) S ∗ [R − S ∗ Y(k − 1)T ] T ( T T ) (40) 2 2 2 2 2 2 2 2 2 478 A. M. E. BAYOUMI AND M. A. RAMADAN where μ is the convergence factor. Substituting from Equations (35) and (36) into Equa- tions (39) and (40) gives T T A F − AX(k − 1)B − CYD B T −1 T −1 X(k) = X(k − 1) + μ(S S ) ∗ ( T T ) 1 1 1 1 E F − EX(k − 1)G − HYN G (41) T −1 T Y(k) = Y(k − 1) + μ(S S ) {[C, H] ∗ [F − AXB − CY(k − 1)D, 2 1 T −1 F − EXG − HY(k − 1)N]} D, N ( T T ) (42) 2 2 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 hierar- chical 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 T T A F − AX(k − 1)B − CY(k − 1)D B T −1 X(k) = X(k − 1) + μ(S S ) ∗ E F − EX(k − 1)G − HY(k − 1)N G T −1 × ( T T ) (43) T −1 Y(k) = Y(k − 1) + μ(S S ) × C, H ∗ F − AX(k − 1)B − CY(k − 1)D,F − EX(k − 1)G − HY(k − 1)N 1 2 T −1 × ( T T ) , (44) D, N μ = or m + n μ = −1 −1 −1 −1 T T T T T T T T λ (S (S S ) S )λ (T (T T ) T ) + λ (S (S S ) S )λ (T (T T ) T ) max 1 1 max 1 1 max 2 2 max 2 2 1 1 1 1 2 2 2 2 (45) In this case, the iterative least-squares solutions of coupled fuzzy Sylvester matrix equations canbewrittenas T T A F − AX(k − 1)B − CY(k − 1)D B T −1 X(k) = X(k − 1) + μ(S S ) ∗ E F − EX(k − 1)G − HY(k − 1)N G T −1 T ) , (46) × ( T T T A F − AX(k − 1)B − CY(k − 1)D B T −1 X(k) = X(k − 1) + μ(S S ) ∗ E F − EX(k − 1)G − HY(k − 1)N G T −1 × ( T T ) , (47) T −1 Y(k) = Y(k − 1) + μ(S S ) C, H ∗ F − AX(k − 1)B − CY(k − 1)D, F − EX(k − 1)G − HY(k − 1)N 1 2 T −1 × ( T T ) , (48) D, N T −1 Y(k) = Y(k − 1) + μ(S S ) 2 FUZZY INFORMATION AND ENGINEERING 479 C, H ∗ F − AX(k − 1)B − CY(k − 1)D, F − EX(k − 1)G − HY(k − 1)N 1 2 T −1 × D, N ( T T ) , (49) μ = or m + n μ = −1 −1 −1 −1 T T T T T T T T λ (S (S S ) S )λ (T (T T ) T ) + λ (S (S S ) S )λ (T (T T ) T ) max 1 1 max 1 1 max 2 2 max 2 2 1 1 1 1 2 2 2 2 We now outline our suggested algorithm. Algorithm IV m×m n×l Step 1 Input crisp matrices A, C, E, H ∈R and B, D, G, N ∈R and input fuzzy matrices m×l F , F ∈R , and number ε. 1 2 m×n Step 2 Given any two initial fuzzy matrices X(0), Y(0) ∈R . Step 3 Compute A B S = , T = , S = C, H , andT = D, N 1 1 2 2 E G Step 4 For k = 1, 2, ··· until convergence T T A F − AX(k − 1)B − CY(k − 1)D B T −1 X(k) = X(k − 1) + μ(S S ) ∗ E F − EX(k − 1)G − HY(k − 1)N G T −1 × ( T T ) , T T A F − AX(k − 1)B − CY(k − 1)D B T −1 X(k) = X(k − 1) + μ(S S ) ∗ E F − EX(k − 1)G − HY(k − 1)N G T −1 × ( T T ) , T −1 Y(k) = Y(k − 1) + μ(S S ) × C, H ∗ F − AX(k − 1)B − CY(k − 1)D, F − EX(k − 1)G − HY(k − 1)N 1 2 T −1 × D, N ( T T ) , T −1 Y(k) = Y(k − 1) + μ(S S ) × C, H ∗ F − AX(k − 1)B − CY(k − 1)D,F − EX(k − 1)G − HY(k − 1)N 1 2 T −1 × D, N ( T T ) , μ = or m + n μ = −1 −1 −1 −1 T T T T T T T T λ (S (S S ) S )λ (T (T T ) T ) + λ (S (S S ) S )λ (T (T T ) T ) max 1 1 max 1 1 max 2 2 max 2 2 1 1 1 1 2 2 2 2 480 A. M. E. BAYOUMI AND M. A. RAMADAN Step 5 If ||X(k) − X(k-1) ||/||X(k)|| <ε, ||X(k) − X(k-1) ||/||X(k)|| <ε, ||Y(k) − Y(k-1) ||/||Y(k)|| <ε and ||Y(k) − Y(k-1) ||/||Y(k)|| <ε stop; otherwise go to step 6. Step 6 Set k = k + 1 and return to step 4. Step 7 End. Theorem 3.2: If the coupled fuzzy Sylvester matrix equations (2) are consistent and has a ∗ ∗ m×n ∗ ∗ m×n ∗ ∗ unique fuzzy solutions X = (X , X ) ∈R and Y = (Y , Y ) ∈R and μ = or m + n μ = −1 −1 −1 −1 T T T T T T T T λ (S (S S ) S )λ (T (T T ) T ) + λ (S (S S ) S )λ (T (T T ) T ) max 1 1 max 1 1 max 2 2 max 2 2 1 1 1 1 2 2 2 2 then the iterative sequence {X(k)}, {X(k)}, {Y(k)} and {Y(k)} generated by algorithm IV con- ∗ ∗ ∗ ∗ ∗ ∗ ∗ verges to X , X , Y and Y , that is, lim X(k) = X ,lim X(k) = X ,lim Y(k) = Y and k→∞ k→∞ k→∞ lim Y(k) = Y for any initial fuzzy matrices X(0) , X(0) , Y(0) and Y(0) . k→∞ Proof: The proof is similar to Theorem 3.1. 3.5. 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 T T B ⊗AD ⊗ C vec (X) vec (F ) T T G ⊗EN ⊗ H vec (Y) vec (F ) + + + + + + + + If A , B , C , D , E , G , H , N contain the positive entries of A, B, C, D, E, G, H, N,respec- − − − − − − − − tively, and A , B , C , D , E , G , H , N contain the absolute value of negative entries of + − + − + − A, B, C, D, E, G, H, N,respectively,itisobviousthat A = A − A , B = B − B , C = C − C , + − + − + − + − + − D = D − D , E = E − E , G = G − G , H = H − H , N = N − N . So, according to the properties of Kronecker operators it can be written as T + − T + − B ⊗ A = (B − B ) ⊗ (A − A ) T T T T + + − − + − − + = (B ⊗ A + B ⊗ A ) − (B ⊗ A + B ⊗ A ) , Similarly T T T T T + + − − + − − + D ⊗ C = (D ⊗ C + D ⊗ C ) − (D ⊗ C + D ⊗ C ) , T T T T T + + − − + − − + G ⊗ E = (G ⊗ E + G ⊗ E ) − (G ⊗ E + G ⊗ E ) , FUZZY INFORMATION AND ENGINEERING 481 T T T T T + + − − + − − + and N ⊗ H = (N ⊗ H + N ⊗ H ) − (N ⊗ H + N ⊗ H ) . T T B ⊗AD ⊗ C Q = T T G ⊗EN ⊗ H T T T T + + − − + + − − B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C Q = T T T T + + − − + + − − G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H T T T T + − − + + − − + B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C T T T T + − − + + − − + G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H Q = S − S (50) 1 2 where T T T T + + − − + + − − B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C S = , T T T T + + − − + + − − G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H T T T T + − − + + − − + B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C S = . T T T T + − − + + − − + G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H In addition, it can be concluded that T T T T + + − − + + − − B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C T = T T T T + + − − + + − − G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H T T T T + − − + + − − + B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C T T T T + − − + + − − + G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H T = S + S . (51) 1 2 Now, the coupled fuzzy Sylvester matrix equations (2) can be written as SM = N, where T T T T + + − − + + − − S S B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C 1 2 S = , S = , T T T T + + − − + + − − S S 2 1 G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H ⎡ ⎤ vec(X) T T T T + − − + + − − + ⎢ ⎥ B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C vec(Y) ⎢ ⎥ S = , M = , T T T T + − − + + − − + ⎣ ⎦ − vec(X) G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H − vec(Y) ⎡ ⎤ vec(F ) vec(F ) N = ⎣ ⎦ − vec(F ) − vec(F ) 2 482 A. M. E. BAYOUMI AND M. A. RAMADAN Now, applying Lemma 2.1 for the matrix equation SM = N, then the following least squares based iterative algorithm leads to lim M(k) = M k→∞ ⎡ ⎤ ⎡ ⎤ vec(X(k)) vec(X(k-1)) ⎢ ⎥ ⎢ ⎥ vec(Y(k)) vec(Y(k-1)) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎣ − vec(X(k))⎦ ⎣ − vec(X(k-1))⎦ − vec(Y(k)) − vec(Y(k-1)) ⎛ ⎡ ⎤ ⎞ ⎡ ⎤ vec(X(k-1)) vec(F ) ⎜ ⎢ ⎥ ⎟ ⎢ ⎥ vec(Y(k-1)) ⎜ vec(F ) S S ⎢ ⎥ ⎟ 2 1 2 T −1 T ⎢ ⎥ + μ(S S) S ⎜ − ⎢ ⎥ ⎟ , ⎣ ⎦ − vec(F ) S S ⎝ ⎣ − vec(X(k-1))⎦ ⎠ 1 2 1 − vec(F ) − vec(Y(k-1)) 0 <μ< 2 S S 1 2 Corollary 3.2: Let matrix S be in the form S = where S S 2 1 T T T T + + − − + + − − B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C S = , T T T T + + − − + + − − G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H T T T T + − − + + − − + B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C S = , T T T T + − − + + − − + G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H then the matrix {1,3} {1,3} {1,3} {1,3} T + Q T − Q {1,3} S = (52) {1,3} {1,3} {1,3} {1,3} 2 T − Q T + Q {1,3} {1,3} is a {1, 3}-inverse of the matrix S, where T and Q are {1, 3}-inverse of the matrices T and Q, respectively. In particular, the Moore–Penrose inverse of the matrix S is † † † † 1 T + Q T − Q S = (53) † † † † T − Q T + Q We now outline our suggested algorithm. Algorithm V m×m n×l Step 1 Input crisp matrices A, C, E, H ∈R and B, D, G, N ∈R and input fuzzy matrices m×l F , F ∈R , and number ε. 1 2 m×n Step 2 Given any two initial fuzzy matrices X(0), Y(0) ∈R . Step 3 Compute T T T T + + − − + + − − S S B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C 1 2 S = , S = , T T T T + + − − + + − − S S 2 1 G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H T T T T + − − + + − − + B ⊗ A + B ⊗ A D ⊗ C + D ⊗ C S = T T T T + − − + + − − + G ⊗ E + G ⊗ E N ⊗ H + N ⊗ H FUZZY INFORMATION AND ENGINEERING 483 + + + + + + + + where A , B , C , D , E , G , H , N contain the positive entries of A, B, C, D, E, G, H, N, − − − − − − − − respectively, and A , B , C , D , E , G , H , N contain the absolute value of negative + − + − + − + entries of A, B, C, D, E, G, H, N where A = A − A , B = B − B , C = C − C , D = D − − + − + − + − + − D , E = E − E , G = G − G , H = H − H , N = N − N . Step 4 For k = 1, 2, ··· until convergence ⎡ ⎤ ⎡ ⎤ vec(X(k)) vec(X(k-1)) ⎢ ⎥ ⎢ ⎥ vec(Y(k)) vec(Y(k-1)) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ − vec(X(k))⎦ ⎣ − vec(X(k-1))⎦ − vec(Y(k)) − vec(Y(k-1)) ⎛ ⎡ ⎤ ⎞ ⎡ ⎤ vec(X(k-1)) vec(F ) ⎜ ⎢ ⎥ ⎟ ⎢ ⎥ vec(Y(k-1)) vec(F ) S S ⎜ 2 1 2 ⎢ ⎥ ⎟ T −1 T ⎢ ⎥ + μ(S S) S − , ⎜ ⎢ ⎥ ⎟ ⎣ ⎦ ⎝ − vec(F ) S S ⎣ − vec(X(k-1))⎦ ⎠ 1 2 1 − vec(F ) − vec(Y(k-1)) 0 <μ< 2 Step 5 If ||X(k) − X(k-1) ||/||X(k)|| <ε, ||X(k) − X(k-1) ||/||X(k)|| <ε, ||Y(k) − Y(k-1) ||/||Y(k)|| <ε and ||Y(k) − Y(k-1) ||/||Y(k)|| <ε stop; otherwise go to step 6. Step 6 Set k = k + 1 and return to step 4. Step 7 End. 4. 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), AX + YB = C, DX + YE = F. Given ⎡ ⎤ ⎡ ⎤ 125 312 13 22 ⎣ ⎦ ⎣ ⎦ A = , D = , B = 341 , E = 413 , 25 13 201 522 (−36 + 38 r,36 − 38 r)(−27 + 30 r,27 − 30 r)(−50 + 33 r,50 − 33 r) C = , (−40 + 38 r,40 − 38 r)(−30 + 24 r,30 − 24 r)(−42 + 27 r,42 − 27 r) (−63 + 54 r,63 − 54 r)(−29 + 21 r,29 − 21 r)(−46 + 34 r,46 − 34 r) F = . (−48 + 33 r,48 − 33 r)(−21 + 13 r,21 − 13 r)(−36 + 21 r,36 − 21 r) 484 A. M. E. BAYOUMI AND M. A. RAMADAN Table 1. The iterative solution for algorithm I for μ = 0.5 kX(k) = (X(k), X(k)) and Y(k) = (Y(k), Y(k)) 1.61 + .238 r - 2.76 + 1.81 r - 4.77 + 1.82 r - .407 - .238 r 2.88 - 1.81 r4.88 - 1.82 r 50 X(k) = X(k) = - 4.28 + 5.20 r - 1.23 + 1.51 r - 2.70 + 2.16 r 4.15 - 5.20 r 1.58 - 1.51 r3.29 - 2.16 r - 3.96 + 2.44 r - 1.66 + 3.82 r - 5.40 + 2.66 r 4.45 - 2.44 r 1.82 - 3.82 r5.66 - 2.66 r Y(k) = , Y(k) = .8978 - .151 r - 2.13 + 1.53 r - 4.78 + 1.41 r - .363 + .151 r 2.54 - 1.53 r4.41 - 1.41 r - .516 + 1.68 r - 2.95 + 1.97 r - 4.95 + 1.97 r .737 - 1.68 r 2.97 - 1.97 r4.97 - 1.97 r 100 X(k) = , X(k) = - 4.05 + 5.03 r - 1.86 + 1.91 r - 3.76 + 2.84 r 4.02 - 5.03 r1.92 - 1.91 r3.87 - 2.84 r - 4.80 + 2.89 r - 1.93 + 3.96 r - 5.90 + 2.95 r 4.89 - 2.89 r 1.96 - 3.96 r5.95 - 2.95 r Y(k) = , Y(k) = - .646 + .810 r - 2.84 + 1.91 r - 4.14 + 1.07 r .811 - .810 r 2.91 - 1.91 r4.07 - 1.07 r - .911 + 1.94 r - 2.99 + 1.99 r - 4.99 + 1.99 r .951 - 1.94 r 2.99 - 1.99 r4.99 - 1.99 r 150 X(k) = , X(k) = - 4.010 + 5.00 r - 1.97 + 1.98 r - 3.95 + 2.97 r 4.00 - 5.00 r1.98 - 1.98 r3.97 - 2.97 r - 4.96 + 2.98 r - 1.98 + 3.99 r - 5.98 + 2.99 r 4.98 - 2.98 r 1.99 - 3.99 r5.99 - 2.99 r Y(k) = , Y(k) = - .935 + .965 r - 2.97 + 1.98 r - 4.02 + 1.01 r .965 - .965 r 2.98 - 1.98 r4.01 - 1.01 r - 1.02 + 2.01 r - 3.00 + 2.00 r - 5.00 + 2.00 r 1.01 - 2.01 r 3.00 - 2.00 r5.00 - 2.00 r 195 X(k) = , X(k) = - 3.99 + 4.99 r - 2.00 + 2.00 r - 4.01 + 3.00 r 3.99 - 4.99 r 2.00 - 2.00 r4.00 - 3.00 r - 5.00 + 3.00 r - 2.00 + 4.00 r - 6.00 + 3.00 r 5.00 - 3.00 r 2.00 - 4.00 r6.00 - 3.00 r Y(k) = , Y(k) = - 1.01 + 1.00 r - 3.00 + 2.00 r - 3.99 + .99 r 1.00 - 1.00 r 3.00 - 2.00 r 3.99 - .99 r This system of coupled fuzzy Sylvester matrix equations has a unique solution (−1 + 2 r,1 − 2 r)(−3 + 2 r,3 − 2 r)(−5 + 2 r,5 − 2 r) X = , (−4 + 5 r,4 − 5 r)(−2 + 2 r,2 − 2 r)(−4 + 3 r,4 − 3 r) (−5 + 3 r,5 − 3 r)(−2 + 4 r,2 − 4 r)(−6 + 3 r,6 − 3 r) Y = . (−1 + r,1 − r)(−3 + 2 r,3 − 2 r)(−4 + r,4 − r) Algorithm I and algorithm II are applied to solve generalized Sylvester matrix equations (1). (1, 1)(1, 1)(1, 1) When selecting the initial matrices as X(0), Y(0) = . Algorithm I is (1, 1)(1, 1)(1, 1) 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 solu- tion 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 giveninTable 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), AX + YB = C, DX + YE = F. FUZZY INFORMATION AND ENGINEERING 485 Table 2. The iterative solution for algorithm II for μ = 0.5. kX(k) = (X(k), X(k)) and Y(k) = (Y(k), Y(k)) - 1.71 + 2.38 r - 3.07 + 2.03 r - 5.06 + 2.03 r 1.38 - 2.38 r 3.03 - 2.03 r5.03 - 2.03 r 50 X(k) = X(k) = - 3.919 + 4.95 r - 2.20 + 2.10 r - 4.34 + 3.18 r 3.95 - 4.95 r2.10 - 2.10 r4.18 - 3.18 r - 4.805 + 2.89 r - 1.93 + 3.96 r - 5.90 + 2.95 r 4.89 - 2.89 r 1.96 - 3.96 r5.95 - 2.95 r Y(k) = , Y(k) = - .646 + .810 r - 2.84 + 1.91 r - 4.14 + 1.07 r .8117 - .810 r2.91 - 1.91 r4.07 - 1.07 r - 1.02 + 2.01 r - 3.00 + 2.00 r - 5.00 + 2.00 r 1.01 - 2.01 r 3.00 - 2.00 r5.00 - 2.00 r 100 X(k) = , X(k) = - 3.99 + 4.99 r - 2.00 + 2.00 r - 4.01 + 3.00 r 3.99 - 4.99 r2.00 - 2.00 r4.00 - 3.00 r - 4.99 + 2.99 r - 1.99 + 3.99 r - 5.99 + 2.99 r 4.99 - 2.99 r1.99 - 3.99 r5.99 - 2.99 r Y(k) = , Y(k) = - .988 + .993 r - 2.99 + 1.99 r - 4.00 + 1.00 r .993 - .993 r2.99 - 1.99 r4.00 - 1.00 r - 1.00 + 2.00 r - 3.00 + 2.00 r - 5.00 + 2.00 r 1.00 - 2.00 r 3.00 - 2.00 r5.00 - 2.00 r 120 X(k) = , X(k) = - 3.99 + 5.00 r - 2.00 + 2.00 r - 4.00 + 3.00 r 4.00 - 5.00 r 2.00 - 2.00 r4.00 - 3.00 r - 4.99 + 2.99 r - 1.99 + 4.00 r - 5.99 + 3.00 r 4.99 - 2.99 r 2.00 - 4.00 r6.00 - 3.00 r Y(k) = , Y(k) = - .99 + .99 r - 2.99 + 1.99 r - 4.00 + 1.00r .99 - .99 r 2.99 - 1.99 r 4.00 - 1.00r Given 2 −1 2 −3 2 −3 3 −1 A = , D = , B = , E = , −33 3 −2 −13 −25 (7 r,18 − 7 r)(−8 + 13 r,13 − 13 r) C = , (−17 + 10 r,15 − 10 r)(−27 + 18 r,6 − 18 r) (−8 + 12 r,23 − 12 r)(−4 + 19 r,27 − 19 r) F = . (−3 + 12 r,32 − 12 r)(−18 + 18 r,23 − 18 r) This system of coupled fuzzy Sylvester matrix equations has a unique solution (1 + r,5 − r)(2 + r,4 − r) (3 + r,4 − r)(1 + 2 r,5 − 2 r) X = , Y = (−1 + r,3 − r)(1 + 2 r,3 − 2 r) (2 + r,3 − r)(−3 + 2 r,3 − 2 r) We use algorithm III to solve generalized Sylvester matrix equations (1). ⎡ ⎤ ⎡ ⎤ 20002000 01000010 ⎢ ⎥ ⎢ ⎥ 03000200 30000001 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 00200030 00013000 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00030003⎥ ⎢ 00300300⎥ S = ⎢ ⎥ , S = ⎢ ⎥ , 1 2 ⎢ 20003000⎥ ⎢ 03000020⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 30000300 02000002 ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ 00200050 00031000 00300005 00020100 † † † † S S T + Q T − Q 1 2 T = S + S , Q = S − S , S = , S = 1 2 1 2 † † † † S S 2 T − Q T + Q 2 1 486 A. M. E. BAYOUMI AND M. A. RAMADAN ⎡ ⎤ ⎡ ⎤ 7r 1 + r ⎢ ⎥ ⎢ ⎥ −1 + r −17 + 10 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 + r −8 + 13 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 + 2 r −27 + 18 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 3 + r ⎢ −8 + 12 r ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −3 + 12 r ⎥ ⎢ 2 + r ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ vec(C) −4 + 19 r 1 + 2 r vec(X) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ vec(Y) vec(F) −18 + 18 r −3 + 2 r † † † ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ M = = S N = S = S = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ − vec(X) − vec(C) −18 + 7 r −5 + r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − vec(Y) − vec(F) −15 + 10 r −3 + r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −13 + 13 r −4 + r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −6 + 18 r ⎥ ⎢ −3 + 2 r⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −23 + 12 r −4 + r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −32 + 12 r −3 + r ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ −27 + 19 r −5 + 2 r −23 + 18 r −3 + 2 r (0, 0)(0, 0) When the initial matrices are chosen as X(0), Y(0) = . Algorithm III is (0, 0)(0, 0) convergent for 0 <μ< 0.5. After 118 iterations we obtain (1.0000 + 1.0000 r, 5.0000 − 1.0000 r)(2.0000 + 1.0000 r, 3.9999 − 0.9999 r) X = , (−1.0000 + 1.0000 r, 3.0000 − 0.9999 r)(0.9999 + 2.0000 r, 2.9999 − 1.9999 r) (2.9999 + 0.9999 r, 3.9999 − 0.9999 r)(0.9999 + 2.0000 r, 5.0000 − 1.9999 r) Y = (1.9999 + 0.9999 r, 3.0000 − 0.9999 r)(−3.0000 + 2.0000 r, 3.0000 − 2.0000 r) Example 4.3: In this example, we illustrate our theoretical results of algorithm V for solving coupled fuzzy Sylvester matrix equations given in (1), AXB + CYD = F , EXG + HYN = F . 1 2 Given 2 −2 −1 −3 21 −23 A = , C = , E = , H = , −34 2 −4 −12 1 −2 32 −12 3 −1 B = , D = , G = , 2 −1 2 −2 0 −1 5 −3 (−47 + 42 r,37 − 33 r)(−58 + 34 r,14 − 30 r) N = , F = , 10 (−37 + 55 r,99 − 71 r)(−51 + 48 r,62 − 57 r) (−17 + 49 r,68 − 42 r)(−40 + 22 r,5 − 29 r) F = (−26 + 27 r,37 − 38 r)(−17 + 21 r,16 − 15 r) This system of coupled fuzzy Sylvester matrix equations has a unique solution (1 + r,3 − r)(1 + 2 r,4 − r) X = , (2 + r,5 − 2 r)(1 + r,3 − r) FUZZY INFORMATION AND ENGINEERING 487 (3 + r,5 − 2 r)(2 + r,3 − r) Y = . (2 + r,4 − r)(−1 + r,3 − r) We apply algorithm V to solve the generalized Sylvester matrix equations (2). ⎡ ⎤ ⎡ ⎤ 604 0 130 0 0604 0 0 26 ⎢ ⎥ ⎢ ⎥ 012080 4 4 0 9060 2 0 08 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 400 2 002 6 0420 2 6 00 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 083 0 400 8 6004 0 8 40 ⎢ ⎥ ⎢ ⎥ S = ⎢ ⎥ , S = ⎢ ⎥ , 1 2 ⎢ 6 3 0 001503⎥ ⎢ 000010 0 20⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 060 0 501 0 3000 0 1002 ⎢ ⎥ ⎢ ⎥ ⎦ ⎦ ⎣ ⎣ 000 0 600 0 2121 0 9 00 101 0 060 0 0202 3 0 00 † † † † S S 1 T + Q T − Q 1 2 T = S + S , Q = S − S , S = , S = 1 2 1 2 † † † † S S T − Q T + Q 2 1 ⎡ ⎤ ⎡ ⎤ −47 + 42 r 0.9999 + 1.0000 r ⎢ ⎥ ⎢ ⎥ −37 + 55 r 2.0000 + 0.9999 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −58 + 34 r 0.9999 + 2.0000 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1.0000 + 0.9999 r −51 + 48 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 3.0000 + 0.9999 r ⎢ −17 + 49 r⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1.9999 + 1.0000 r ⎥ ⎡ ⎤ −26 + 27 r ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ vec(F ) ⎢ ⎥ ⎢ ⎥ 1.9999 + 1.0000 r vec(X) −40 + 22 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ vec(F ) −0.9999 + 0.9999 r vec(Y) 2 −17 + 21 r † † † ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ M = = S N = S = S = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ −37 + 33 r −3.000 + 1.0000 r − vec(X) − vec(F ) 1 ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ −99 + 71 r −4.9999 + 1.9999 r − vec(Y) ⎢ ⎥ ⎢ ⎥ − vec(F ) ⎢ ⎥ ⎢ ⎥ −14 + 30 r −4.0000 + 1.0000 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −62 + 57 r⎥ ⎢ −2.9999 + 0.9999 r⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −68 + 42 r −4.9999 + 1.9999 r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −37 + 38 r −4.0000 + 1.0000 r ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ −5 + 29 r −3.000 + 1.0000 r −16 + 15 r −2.9999 + 0.9999 r (0, 0)(0, 0) When the initial matrices are chosen as X(0), Y(0) = .ThealgorithmVis (0, 0)(0, 0) convergent for 0 <μ< 0.7. After 145 iterations we obtain (1.0000 + 0.9999 r, 3.0000 − 1.0000 r)(0.9999 + 2.0000 r, 4.0000 − 1.0000 r) X = , (2.0000 + 0.9999 r, 4.9999 − 1.9999 r)(1.0000 + 0.9999 r, 2.9999 − 0.9999 r) (3.0000 + 0.9999 r, 4.9999 − 1.9999 r)(1.9999 + 1.0000 r, 3.0000 − 1.0000 r) Y = , (1.9999 + 1.0000 r, 4.0000 − 1.0000 r)(−0.9999 + 0.9999 r, 2.9999 − 0.9999 r) 5. 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 488 A. M. E. BAYOUMI AND M. A. RAMADAN 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 cou- pled 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 itera- tive 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. Acknowledgments The authors would like to express their heartfelt thanks to the editor and anonymous referees for their useful comments. Conflict of interest The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding This research received no specific grant from any funding agency in the public, commercial, or not- for-profit sectors. ORCID Ahmed M. E. 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Journal

Fuzzy Information and Engineering – Taylor & Francis

Published: Oct 1, 2020

Keywords: Coupled fuzzy Sylvester matrix equations; Iterative algorithm; Kronecker product; Frobenius norm; Star product

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