ELA INTERVAL SYSTEM OF MATRIX EQUATIONS WITH TWO UNKNOWN MATRICES

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1 INTERVAL SYSTEM OF MATRIX EQUATIONS WITH TWO UNKNOWN MATRICES A RIVAZ, M MOHSENI MOGHADAM, AND S ZANGOEI ZADEH Abstract In this paper, we consider an interval system of matrix equations contained two equations with two unknown matrices as A11 X +YA 12 = C 1, A 21 X +YA 22 = C 2 We define a solution set for this system and then we study some conditions that the solution set is bounded Finally, we present a direct method and an iterative method for solving this interval system Key words Interval matrix, Interval linear systems, System of matrix equations, Solution set AMS subject classifications 65F30 1 Introduction Matrix equations and systems of matrix equations have many applications in sciences and engineering, such as electromagnetic scattering, structural mechanics and computation of the frequency response matrix in control theory A sample of these systems of matrix equations is as the following: A11 X +YA 12 = C 1, A 21 X +YA 22 = C 2, (11) where A i1, A i2 and C i, for i = 1,2, are the known real matrices of dimensions m m, n n and m n, respectively, while the unknown matrices X and Y are m n real matrices The general state of this problem can be seen in 1 The elements ofa ij, for i,j = 1,2, occurringin practice areusually obtainedfrom experiments, hence they may appear with uncertainties We represent the uncertain elements in interval forms Therefore, with the existence of uncertainties in data, the system of matrix equations (11) is transformed to the following interval system of Received by the editorsondecember 2, 2013 Accepted forpublication onjune 2, 2014 Handling Editor: Michael Tsatsomeros Mahani Mathematical Research Center, University of Kerman, Kerman, Iran (arivaz@ukacir) This work has been supported by the Mahani Mathematical Research Center of Shahid Bahonar University of Kerman Mathematics Department, Islamic Azad University of Kerman, Kerman, Iran (mohseni@ukacir) Department of Mathematics,University of Kerman, Kerman, Iran (szangoeizadeh@gmailcom) 478

2 matrix equations Interval System of Matrix Equations With Two Unknown Matrices 479 A11 X +YA 12 = C 1, A 21 X +YA 22 = C 2, (12) where A ij and C i, for i,j = 1,2, are interval matrices Note that bold-face letters are used to show intervals Some samples of interval matrix equations such as AX = B and the interval Sylvester equation AX + XB = C, have been considered previously; see 3, 16 In this paper, we use notations R and R m n as the field of real numbers and the vector space of m n real matrices, respectively We denote the set of all m n interval matrices by IR m n For the interval matrix A = A,A, the center matrix denoted by radius matrix denoted by rad(a) are respectively defined as Ǎ and the Ǎ = 1 2 (A+A) and rad(a) = 1 2 (A A) It is clear that A = Ǎ rad(a),ǎ+rad(a) We assume that the reader is familiar with a basic interval arithmetic and interval operatorsontheintervalmatrices Formoredetails,wereferto7,8 Ann ninterval matrix A = A,A is said to be regular if each A A is nonsingular For two interval matrices A IR m n and B IR k t, the Kronecker product denoted by is defined by the mk nt block interval matrix a 11 B a 1n B A B =, a m1 B a mn B and vec(a) is defined as an mn-interval vector and obtained by stacking the columns of A, ie, vec(a) = (A 1,A 2,,A n ) T, where A j is the j th column of A Note that these definitions are similar to those of real matrices Theorem 11 If A,C R m m, B,D R n n and X R m n, then we have 1 (A+C) B = (A B)+(C B), 2 B (A+C) = (B A)+(B C), 3 (A B)(C D) = AC BD, 4 vec(ax) = (I n A)vec(X),

3 480 A Rivaz, M Mohseni Moghadam, and S Zangoei Zadeh 5 vec(xb) = (B T I m )vec(x), 6 λ(a B) = λ(a)λ(b), 7 λ((i n A)+(B I m )) = λ(a)+λ(b), where λ(a) is an eigenvalue of A For discussion of convergence in the context of interval analysis, the distance between two intervals x = x,x and y = y,y is denoted by d(x,y) and is defined as the following: d(x,y) = max x y, x y } For more information, see in 7 Let x k } be a sequence of intervals We say that x k } is convergent if there exists an interval x such that for every ǫ > 0, there is a natural number N = N(ǫ) such that d(x k,x ) < ǫ whenever k > N A necessary and sufficient condition for convergence of sequence x k } is stated in the next theorem Theorem 12 The interval sequence x k } is convergent to x = x,x if and only if x k x and x k x in the sense of real sequences 2 Main results Consider the interval system of matrix equations (12) We define the solution set for the system (12) by Σ(X,Y) = (X,Y) : X,Y R m n },A i1 X +YA i2 = C i ; i = 1,2 (21) for some A i1 A i1,a i2 A i2,c i C i Similar to solving of interval linear systems 8, 13, the solution set of an interval system is generally of a complicated structure But if Σ(X,Y) is bounded, we look for an enclosure of this set, ie, for a pair of interval matrices (X,Y) satisfying Σ(X,Y) (X,Y) In the following two subsections, we will present a direct method to obtain an enclosure of Σ(X,Y) and conditions under which Σ(X,Y) is bounded Also we will present an iterative algorithm to solve this important problem The description of a superset of Σ(X,Y) given in the next theorem is similar to that which appeared in the pioneering works of Oettli and Prager 9 for interval linear systems Theorem 21 The solution set Σ(X, Y) defined by (21) satisfies Σ(X,Y) (X,Y) : } Ǎi1X +YǍi2 Či rad(a i1 ) X + Y rad(a i2 )+rad(c i ) ; i = 1,2 (22)

4 Interval System of Matrix Equations With Two Unknown Matrices 481 Proof Let (X,Y) Σ(X,Y), then for some A ij A ij and C i C i for i,j = 1,2, we have Therefore, we have: A i1 X +YA i2 C i = 0 Ǎi1X +YǍi2 Či = Ǎi1 +YǍi2 Či A i1 X YA i2 +C i Ǎi1 A i1 X + Y Ǎi2 A i2 + Či C i rad(a i1 ) X + Y rad(a i2 )+rad(c i ) So the proof is completed 21 Direct method For obtaining an enclosure (X,Y) of Σ(X,Y), suppose that (X,Y) Σ(X,Y) Then by using Theorem 21, we have (Ǎ11)X rad(a 11 ) X +Y(Ǎ12) Y rad(a 12 ) C 1, (Ǎ11)X +rad(a 11 ) X +Y(Ǎ12)+ Y rad(a 12 ) C 1, (Ǎ21)X rad(a 21 ) X +Y(Ǎ22) Y rad(a 22 ) C 2, (Ǎ21)X +rad(a 21 ) X +Y(Ǎ22)+ Y rad(a 22 ) C 2 For given matrices X and Y, we write X = T X and Y = S Y, where denotes the so-called Hadamard product and the matrices T and S are sign matrices of X and Y, respectively In order to find an element x ij for fixed i and j of the interval matrix X = (x ij ), we have to solve 2 2mn linear programming problems as the following: minx ij subject to: (Ǎ11)X rad(a 11 )(T X)+Y(Ǎ12) (S Y)rad(A 12 ) C 1, (Ǎ11)X +rad(a 11 )(T X)+Y(Ǎ12)+(S Y)rad(A 12 ) C 1, (Ǎ21)X rad(a 21 )(T X)+Y(Ǎ22) (S Y)rad(A 22 ) C 2, (Ǎ21)X +rad(a 21 )(T X)+Y(Ǎ22)+(S Y)rad(A 22 ) C 2, and for all possible sign matrices T and matrices S Hence, we need 4mn 2 2mn linear programming problems for finding the interval matrices X and Y, which makes the problem very troublesome even with small values of m and n Accordingly, we tryto find an enclosure(x,y) ofσ(x,y) by aneasiertechnique It can be shown that for any A ij A ij and C i C i, the system of matrix equations A11 X +YA 12 = C 1, A 21 X +YA 22 = C 2

5 482 A Rivaz, M Mohseni Moghadam, and S Zangoei Zadeh can be transformed to the following form Gz = d, where In A 11 A T 12 G = I m I n A 21 A T 22 I m vec(c1 ), d = vec(c 2 ) vec(x), z = vec(y) Now, we consider interval linear system of equations where G IR 2mn 2mn and d IR 2mn are as the following: In A G = 11 A T 12 I m I n A 21 A T 22 I m Gz = d, (23) vec(c1 ), d = vec(c 2 ) We assume that Γ is the solution set of (23) We define It is clear that Θ Γ Θ = vec(x) vec(y) } : (X,Y) Σ(X,Y) (24) Therefore, by solving interval linear system (23) and finding interval vector z as an enclosure of its solution set, we can specify the columns of the interval matrices X and Y which (X,Y) is an enclosure of Σ(X,Y) To solve the interval linear system (23), see 2, 4, 7, 8, 11, 13, 14, 15 As was mentioned before, the enclosure of Σ(X,Y) is achievable if Σ(X,Y) is a bounded set The following theorem represent a condition in order that the solution set of the interval system (12) to be bounded Theorem 22 For all interval m n matrices C 1 and C 2 the solution set defined by (21) is bounded if the system of inequalities Ǎ11X +YǍT 12 rad(a 11 ) X + Y rad(a T 12), (25) Ǎ21X +YǍT 22 rad(a 21) X + Y rad(a T 22 ), (26) have only the trivial solution (X,Y) = (0,0) Proof It is clear that Σ(X,Y) is bounded if and only if the set Θ defined by (24) is bounded Since we have Θ Γ, So it is sufficient to prove that Γ is bounded for

6 Interval System of Matrix Equations With Two Unknown Matrices 483 each 2mn-vector d The inequalities (25) and (26) are equivalent to the following inequalities vec(ǎ11x)+vec(yǎt 12) vec(rad(a 11 ) X )+vec( Y rad(a T 12)), vec(ǎ21x)+vec(yǎt 22 ) vec(rad(a 21) X )+vec( Y rad(a T 22 )) Based on Theorem 11, we can rewrite the above inequalities as (I n Ǎ11)vec(X)+(ǍT 12 I m )vec(y) (I n rad(a 11 )) vec(x) +(rad(a T 12 ) I m)( vec(y) ), (27) (I n Ǎ21)vec(X)+(ǍT 22 I m)vec(y) (I n rad(a 21 )) vec(x) +(rad(a T 22) I m )( vec(y) ) (28) So, if the inequalities (25) and (26) have trivial solution then the inequalities (27) and (28) also have the trivial solution As for the interval system (23), and the inequalities (27) and (28), the inequality Ǧz rad(g) z has only the trivial solution z = 0 Thus, from the result of 12, the solution set Γ is bounded for each 2mn-vector d In the next theorem, we give a necessary and sufficient condition for boundedness of Σ(X,Y) Theorem 23 Suppose that A 11 in the interval system of matrix equations is regular Then for all interval matrices C 1 and C 2, the solution set of (12) is bounded if and only if (A T 22 A 11) (A T 12 A 21) is nonsingular for each A ij A ij, i,j = 1,2 Proof The solution set of interval system of matrix equations (12) is bounded if and only if for all fixed A ij A ij, i,j = 1,2, the matrix G defined by In A 11 A T 12 G = I m I n A 21 A T 22 I, m is nonsingular, or equivalently det(g) 0 Since A 11 is a nonsingular matrix and A 22 is a square matrix, from 6, p 46, it follows that det(g) = det(i n A 11 )det ( (A T 22 I m ) (I n A 21 )(I n A 1 11 )(AT 12 I m ) ) = det(i n A 11 )det ( (A T 22 I m) (A T 12 A 21A 1 11 )) Hence, det(g) 0 if and only if det ( (A T 22 I m ) (A T 12 A 21 A 1 11 )) 0,

7 484 A Rivaz, M Mohseni Moghadam, and S Zangoei Zadeh which is equivalent to det ( (A T 22 A 11) (A T 12 A 21) ) 0 (29) Therefore, the solution set of the interval system of matrix equations (12) is bounded if and only if for all fixed A ij A ij, i,j = 1,2, det ( (A T 22 A 11 ) (A T 12 A 21 ) ) 0 The following corollary is an immediate consequence of the theorem Corollary 24 Suppose that A 11 in the interval system of matrix equations is regular Then for all interval matrices C 1 and C 2, the solution set of (12) is bounded if ( (A T 22 A 11) (A T 12 A 21) ) is regular For checking the regularity of interval matrices, see 5, 10 If we choose A 11 = A 12 = I, then we can state the following result Corollary 25 In the interval system of matrix equations (12), assume that A 11 = A 12 = I Then the corresponding solution set is bounded if and only if where σ(a 21 ) σ(a 21 ) =, σ(a) = λ C : x 0, Ax = λx for some A A} Proof Consider A 11 = I m and A 12 = I n Similar to the proof of previous theorem, the solution set is bounded if and only if for each A 21 A 21 and A 22 A 22 det ( (A T 22 I m) (I n A 21 ) ) 0, (210) The relation (210) is equivalent to that λ i µ j 0 for every eigenvalue λ i, for i = 1,,n, of A 22 and every eigenvalue µ j, for j = 1,,m of A 21 This implies that the solution set is bounded if and only if σ(a 21 ) σ(a 21 ) = Example 26 Consider the interval system of matrix equations A11 X +YA 12 = C 1, A 21 X +YA 22 = C 2,

8 in which A 11 = A 21 = Interval System of Matrix Equations With Two Unknown Matrices 485 1,2 2,25 2, 1 5,6 01,03 0,05 04, 03 02,02, A 12 =, A 22 = 01211, , , , ,03 2,25 05,02 01,03 3,4 4, 3 1,2 6,7, C 1 =, C 2 = 3,4 1,0 1,1 6,8 6,7 1,3 8,9 6,8 Here, ( (A T 22 A 11) (A T 12 A 21) ) is regular The proposed direct method shows that the enclosureofthe solution set is a pairof interval matricesof (X Di,Y Di ) where X Di =, Y Di = 04094, , , ,32972, 22 Iterative method In this subsection, we present an iterative method for solving the interval system of matrix equations (12) Moreover, we discuss the condition of convergence for this method Let us rewrite (12) as the following: A11 X = C 1 YA 12, YA 22 = C 2 A 21 X Assume that an initial guess Y 0 is given Now we define the following iteration equations: A11 X k+1 = C 1 Y k A 12, k 0 (211) Y k+1 A 22 = C 2 A 21 X k+1, TofindtheintervalmatricesX k+1 andy k+1 in(211), weneedtosolveaninterval matrix equation such as AX = B To this end, we consider an interval linear system of the form AX j = B j, where X j and B j are j th columns of X and B, respectively Having considered them, we introduce the following algorithm Algorithm 27 Step 1 Choose Y 0 and set k = 0 Step 2 Solve the interval matrix equations A 11 X k+1 = C 1 Y k A 12, A T 22 Y T k+1 = C 2 T X T k+1 AT 21 and find the interval matrices X k+1 and Y k+1

9 486 A Rivaz, M Mohseni Moghadam, and S Zangoei Zadeh Step 3 If max X k+1 X k, X k+1 X k, Y k+1 Y k, Y k+1 Y k } < ǫ, stop Otherwise, set X k = X k+1, Y k = Y k+1 if k = 0, (212) X k = X k+1 X k, Y k = Y k+1 Y k if k 1 (213) Set k = k +1, and go to the Step 2 As discussed in 7, The relation (213) implies that the algorithm converges in a finite number of iterations By considering a particular condition, the next theorem expresses that the above algorithm converges in the absence of the relation (213) Theorem 28 In the interval system (12), suppose A 11 and A 22 are regular matrices If A 21 A 1 11 A 12A 1 22 < 1 for each A ij A ij, i,j = 1,2, then the algorithm described above converges Proof From (211), we write A11 X k+1 = C 1 Y k A 12, Y k+1 A 22 = C 2 A 21 X k+1 (214) for each A ij A ij and C i C i, i,j = 1,2 Using the relations (214) together with the regularity of A 11 and A 22 yields Y k+1 = C 2 A 1 22 A 21A 1 11 C 1A A 21A 1 11 Y ka 12 A 1 22 (215) Suppose (X,Y ) is the exact solution of (214) So, Y satisfies Y = C 2 A 1 22 A 21A 1 11 C 1A A 21A 1 11 Y A 12 A 1 22 (216) If we apply (215) and (216), then we conclude that Hence, Y k+1 Y = A 21 A 1 11 (Y k Y )A 12 A 1 22 = (A 21 A 1 11 )2 (Y k 1 Y )(A 12 A 1 22 )2 = (A 21 A 1 11 )k+1 (Y 0 Y )(A 12 A 1 22 )k+1 Y k+1 Y A 21 A 1 11 k+1 Y 0 Y A 12 A 1 22 k+1

10 Interval System of Matrix Equations With Two Unknown Matrices 487 This shows that if A 21 A 1 11 A 12A22 1 < 1, then each sequence of the real matrices Y k } constructed by our algorithm converges, which implies the convergence of Y k } Again, from (214), we have It follows that X k+1 = A 1 11 (C 1 Y K A 12 ) X k+1 X = A 1 11 (Y k Y )A 12 So, convergence of Y k } implies convergence of X k } and we can conclude that X k } converges Example 29 Consider the interval system of matrix equations in Example 1,1 1,1 (26) With Y 0 = 1,1 1,1, by performing 4 iterations of the above algorithm we have: X It = 04333, , , ,09067, Y It = 04887, , , ,29937 With ignoring (213) we obtain the following wider interval matrices X It =, Ỹ It = 02549, , , , , , , ,30626 From the result of examples (26) and (29), we conclude that: (X It,Y It ) ( X It,Ỹ It ) (X Di Y Di ) Remark 210 If the condition of the above theorem is not established, we may use the intersection of (X Di,Y Di ) and (X It,Y It ) for obtaining a sharper solution Acknowledgement The authors wish to thank the referee for comments that resulted in improvement of the text of this paper REFERENCES 1 M Dehghan and M Hajarian The general coupled matrix equations over generalized bisymmetric matrices Linear Algebra and its Applications, 432: , J Garloff Pivot tightening for direct methods for solving symmetric positive definite systems of linear interval equations Computing, 94:97 107, B Hashemi and M Dehghan Results concerning interval linear systems with multiple righthand sides and the interval matrix equation AX = B Journal of Computational and Applied Mathematics, 235: , 2011

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