SOLVING FUZZY LINEAR SYSTEMS BY USING THE SCHUR COMPLEMENT WHEN COEFFICIENT MATRIX IS AN M-MATRIX

Transcription

1 Iranian Journal of Fuzzy Systems Vol 5, No 3, 2008 pp SOLVING FUZZY LINEAR SYSTEMS BY USING THE SCHUR COMPLEMENT WHEN COEFFICIENT MATRIX IS AN M-MATRIX M S HASHEMI, M K MIRNIA AND S SHAHMORAD Abstract This paper analyzes a linear system of equations when the righthand side is a fuzzy vector and the coefficient matrix is a crisp M-matrix The fuzzy linear system FLS is converted to the equivalent crisp system with coefficient matrix of dimension 2n 2n However, solving this crisp system is difficult for large n because of dimensionality problems It is shown that this difficulty may be avoided by computing the inverse of an n n matrix instead of Z 1 1 Introduction n n fuzzy linear systems have been studied by many authors [1, 3, 5, 6, 7, 9, 10, 11, 12] In [7] Friedman et al proposed a general model for solving such systems by the embedding approach and stated conditions for the existence of a unique fuzzy solution to n n linear systems A minimum norm solution of the fuzzy under determined systems has been obtained in [10] In this paper we investigate the case where the coefficient matrix in a fuzzy linear system is an M-matrix We first study the properties of this system and then propose a solution using the Schur complement Section 2 provides preliminaries for fuzzy numbers and fuzzy linear systems Several M-matrix properties and the Schur complement formula are also stated The relationship between an M-matrix and its crisp matrix and the existence and expression of the solution to the fuzzy linear system using the Schur complement are discussed in section 3 Numerical examples to illustrate previous sections are given in Section 4 2 Preliminaries 21 Fuzzy Numbers and Fuzzy Linear Systems In this section we recall the basic notion of fuzzy numbers arithmetic and fuzzy linear system Definition 21 [7] A fuzzy number in parametric form is an ordered pair of functions ur, ur, 0 r 1 which satisfy the following conditions: 1 ur is a bounded left continuous nondecreasing function on [0,1] Received: May 2007; Revised: September 2007 and January 2008; Accepted: February 2008 Key words and phrases: Fuzzy linear system, Schur complement, M-matrix, H-matrix

2 16 M S Hashemi, M K Mirnia and S Shahmorad 2 ur is a bounded left continuous nonincreasing function on [0,1] 3 ur ur, 0 r 1 If ur ur ψ, 0 r 1 then ψ is a crisp number A popular representation for fuzzy number is the trapezoidal representation u x 0, y 0, α, β with defuzzifier interval [x 0, y 0 ], left fuzziness α and right fuzziness β The membership function of this trapezoidal number is as follows: 1 α x x 0 + α, x 0 α x x 0 1, x [x 0, y 0 ] ux 1 β y 0 x + β, y 0 x y 0 + β 0, otherwise The parametric form of the number is ur x 0 α + αr, ur y 0 + β βr The operations of addition and scalar multiplication for arbitrary fuzzy numbers, u u, u and v v, v and λ R 1 are defined by u + vr ur + vr, u + vr ur + vr, λur λur, λur λur, λ 0, λur λur, λur λur, λ 0 A general fuzzy linear system is as follows: 1 a 11 x 1 + a 12 x a 1n x n y 1 a 21 x 1 + a 22 x a 2n x n y 2 2 a n1 x 1 + a n2 x a nn x n y n, where the coefficient matrix A a ij, 1 i, j n is a crisp n n matrix and y y i, 1 i n is a fuzzy vector This system is called a fuzzy linear system FLS By 21, the system 21 is equivalent to the following parametric system: n j1 a ijx j n j1 a ijx j y i n j1 a ijx j 3 n j1 a ijx j y i If for a particular i : a ij > 0, 1 j n, then we obviously get n n a ij x j y i, a ij x j y i j1 j1

3 Solving Fuzzy Linear Systems by Using the Schur Complement 17 In general, an arbitrary equation for either y i or y i may include a linear combination of x j s and x j s Consequently, in order to solve the system given by Eq?? one must solve a 2n 2n crisp linear system, where the column on the right is the vector y 1, y 2,, y n, y 1, y 2,, y n T Let us now rearrange the linear system of Eqs?? so that the unknowns are x i, x i, 1 i n and the column on the right is Y y 1, y 2,, y n, y 1, y 2,, y n T Define a 2n 2n matrix Z z ij as follows: a ij 0 z ij a ij, z i+n,j+n a ij, a ij < 0 z i,j+n a ij, z i+n,j a ij 4 Any z ij which is not determined by 21 is zero Then the system?? may be written in the following crisp block form: B 0 C 0 X Y ZX Y 5 C 0 B 0 X Y where B, C R n,n and x 1 r X, X x n r x 1 r x n r, Y y 1 r y n r, Y y 1 r y n r The definition of Z z ij implies that B contains the positive entries of A, C contains the absolute values of the negative entries of A and A B C 22 M-matrices and Some of their Properties Definition 22 [2] A real square matrix A a ij is called an M-matrix if a ij 0; i j and A 1 0 For any real or complex n n matrix A a ij, the comparison matrix, denoted by ΞA ξ ij A, and defined by { aij, i j ξ ij A 6 a ij, i j is called an H matrix if ΞA is an M-matrix Definition 23 [2] A square matrix A is said to be generalized diagonally dominant if a ii x i a ij x j, i 1, 2,, n 7 j i

4 18 M S Hashemi, M K Mirnia and S Shahmorad for some positive vector x x 1, x 2,, x n T, generalized strictly diagonally dominant if 23 is valid with strict inequality and strictly diagonally dominant if 23 is valid for x1, 1,, 1 T Lemma 24 [2] Let A a ij be an n n matrix, with a ij 0, i j and a ii > 0 If A is strictly diagonally dominant, then A is an M-matrix Lemma 25 [2] a A matrix A a ij, with a ij 0, i j, is an M-matrix if and only if there exists a positive vector x, such that Ax is positive b A matrix A a ij is an H-matrix if and only if A is generalized strictly diagonally dominant Theorem 26 [2] Let A a ij be a matrix of order n, with a ij 0, the following are equivalent: a A is generalized strictly diagonally dominant and a ii 0 b A is an M-matrix i j Then 23 Schur Complements Let the matrix A be partitioned into the two-by-two block form A11 A A 12 8, A 21 A 22 where A ii, i 1, 2 are square matrices Definition 27 [2] If A 11 is nonsingular, we define S A/A 11 A 22 A 21 A 1 11 A 12, 9 S is called the Schur complement of A with respect to A 11 Note that S is in the position A 22 of A after we use Gaussian-elimination to convert A 21 to zero in A In fact, the block matrix triangular factorization of A is readily found to be A I 0 A 21 A 1 11 I A S I A 1 11 A 12 0 I 10 Hence we may write the inverse of A in one of the two forms 11 and 12 as follows: A 1 I A 1 11 A 12 A I I 0 S 1 A 21 A 1 11 I A 1 A A 1 11 A 12S 1 A 21 A 1 11 A 1 11 A 12S 1 S 1 A 21 A 1 11 S 1 12

5 If both A 1 11 where Solving Fuzzy Linear Systems by Using the Schur Complement 19 and A 1 22 exist, an alternative expression for A 1 is A 1 S 1 1 A 1 11 A 12S2 1 S2 1 A 21A 1 11 S2 1 S i A/A jj, i j, i, j 1, 2 Theorem 28 [2] Let A be an M-matrix that is partitioned in a two-by-two block form Then the Schur complement exists and is also an M-matrix S A/A 11 A 22 A 21 A 1 11 A 12 3 Solving a Fuzzy Linear System when A Is an M-matrix 31 Solution to a Fuzzy Linear System To study the solution to a fuzzy linear system, it is first necessary to discuss the generalized inverses of the matrix Z in a special structure Definition 31 [5] Let X { x i r, x i r; 1 i n } denote a solution of ZX Y The fuzzy number vector U { u i r, u i r; 1 i n } defined by u i r min { x i r, x i r, x i 1, x i 1 } 13 u i r max { x i r, x i r, x i 1, x i 1 } 14 is called a fuzzy solution of ZX Y If x i r, x i r; 1 i n, are all fuzzy numbers and u i r x i r, u i r x i r, 1 i n, then U is called a strong fuzzy solution Otherwise, U is a weak fuzzy solution From 21 we have the following assertion: Theorem 32 [12] Let matrix Z be in the form 21 Then the matrix Z 1 B + C + B C B + C B C 2 B + C B C B + C + B C 15 is a g-inverse of the matrix Z, where B + C and B C are g-inverses of the matrices B + C and B C, respectively In particular, the Moore-Penrose inverse of the matrix Z is Z 1 B + C + B C B + C B C 2 B + C B C B + C + B C 16 In this paper, we consider only the case when m n; thus Z is nonsingular if and only if the matrices A B C and B + C are both nonsingular [7, Theorem 1] Hence, as a direct consequence of Theorem 32, we have the following corollary

6 20 M S Hashemi, M K Mirnia and S Shahmorad Corollary 33 [7] If Z 1 exists, it must have the same structure as Z, ie Z 1 T1 T 2 T 2 T 1 where T [B + C 1 + B C 1 ], T [B + C 1 B C 1 ] The next result provides a sufficient condition for the solution vector to be a fuzzy vector Theorem 34 [12] For the consistent system 21 and any g-inverse Z of the coefficient matrix Z, X Z Y is a solution to the system and therefore it admits a weak or strong fuzzy solution In particular, if Z is nonnegative with the special structure 32, then X Z Y admits a strong fuzzy solution for arbitrary fuzzy vector Y Corollary 35 [12] For the n n fuzzy linear system, if Z 1 exists, then for an arbitrary fuzzy vector Y, the unique solution X Z 1 Y is a fuzzy vector if Z 1 is nonnegative 32 Properties of the Equivalent Crisp Matrix Z We consider the solution of FLS, Ax ỹ when A is an M-matrix and ỹ is a fuzzy vector Then 21, B C implies that z ij 0, 1 i, j n and Z where B is the diagonal C B matrix containing the diagonal entries of A and C contains the absolute value of the off-diagonal entries of A In other words, B diaga 11, a 22,, a nn, and so B 1 diag a 11, a 22,, a nn, and C c ij n n where { cii 0, 1 i n c ij a ij, 1 i, j n, i j Theorem 36 If A is an M-matrix then Z, as defined by 21, is an H-matrix Proof From theorem 26, it is clear that A is generalized strictly diagonally dominant and a ii 0 Thus from lemma 25 for some x > 0, x R n we have a ii x i > j i a ij x j, i 1, 2,, n 17 To complete the proof, it is sufficient to show that Z is generalized strictly diagonally dominant Let y [x T, x T ], where x is the vector corresponding to the matrix A Then it may easily be checked that the relation z ii y i > 2n j1 holds for the matrix Z defined in 21 z ij y j, j i, 1 i 2n 18

7 Solving Fuzzy Linear Systems by Using the Schur Complement Solution Using the Schur Complement In this section we employ the Schur complement for computing Z 1 where Z is defined by 21 with block form B C Z where B is a diagonal matrix and C is a nondiagonal matrix whose C B entries are absolute values of the off-diagonal entries of A if A is an M-matrix It is clear from 27, 23 and 21 that S S 1 S 2 B CB 1 C and so Z 1 S 1 B 1 CS 1 19 S 1 CB 1 S 1 Theorem 37 Let A be an M-matrix, in FLS, Z be the extended crisp matrix of A, and S be its Schur complement Then we have S 1 CB 1 B 1 CS 1 Proof Let P S 1, Q 1 B 1 CS 1, Q 2 S 1 CB 1 We must prove Q 1 Q 2 From 33 and 21 we have ZZ 1 B C P Q1 I 0 I, or, C B P 0 I thus { BQ1 + CP 0 CP + BQ 2 0 which completes the proof From theorem 37 we have where P S 1, or Q 2 { Q1 B 1 CP Q 2 B 1 CP Z 1 P Q Q P }, 20 Q P CB 1 Hence 21 and 33 imply that X P Q Y X Q P Y X P Y QY X P Y QY 34 Some Properties of Z 1 To investigate properties of Z 1, we shall need the following definitions and lemmas: Definition 38 [4] Let A a ij be an m n matrix We define a matrix norm A as follows: A max 1 i m n j1 a ij Lemma 39 [4] For any matrices, A and B, we have and consequently A 2 A 2 AB A B, 21

8 22 M S Hashemi, M K Mirnia and S Shahmorad Lemma 310 [4] The matrix series converges to I A 1 if A < 1 I + A + A A k + Theorem 311 Let A be a strictly diagonally dominant matrix with the extended crisp matrix defined by 21 and the inverse defined by by 33 Then P 0, Q 0 Proof We know that P S 1 B CB 1 C 1, thus B 1 S I B 1 CB 1 C Let E B 1 CB 1 C We prove E < 1 From 21 we have 1 a a 12 a 1n 1 B 1 0 a 22 0 a 21 0 a 2n C a nn a n1 a n2 0 a 0 12 a a 11 1n 11 a 21 a a n a 22 a n1 a n2 a nn a nn 0 Thus by the Lemma 39 and inequalities 32 we obtain Therefore and from lemma 310 B 1 C max 1 i n n j1 a ij a ii < 1 E B 1 C 2 B 1 C 2 < 1, I E 1 I + E + E Hence S 1 B I E 1 0 and P S 1 S 1 BB 1 0 From Q P CB 1 and P 0, C 0, B 1 0, it is clear that Q 0 and the proof is complete Theorem 312 Let A a ij be an n n matrix with the following properties: a a ij 0 for i j and there exists at least one negative off-diagonal entry b a ii > 0, i 1,, n c A is strictly diagonally dominant X Then there exists a fuzzy vector Y such that X Z 1 Y is not a fuzzy X vector ie x 1 r x 1 r, x 1 r x n r x n r, x n r

9 Solving Fuzzy Linear Systems by Using the Schur Complement 23 is not a strong fuzzy solution for system 21 Note: If the assumption a fails then C 0 and there is a fuzzy solution by corollary 35 Proof Suppose p 11 p 1n P p n1 p nn, Q q 11 q 1n q n1 q nn From the assumptions, Theorem 311 holds and from second part of assumption a,q 0 Since P is non-singular and non-negative, hence there exist i 0 and j 1 such that q i0,j 1 < 0 and p i0,j 0 > 0, 1 i 0, j 0, j 1 n Now we construct a fuzzy vector Y such that X Z 1 Y is a non-fuzzy vector Let us write p 11 p 1n q 11 q 1n y 1 r X Z 1 p n1 p nn q n1 q nn y n r Y q 11 q 1n p 11 p 1n y 1 r q n1 q nn p n1 p nn y n r x 1 r x n r x 1 r 22 x n r It is sufficient to construct Y such that x i0 r is a non-fuzzy number From 34 we have x i0 r p i01y 1 r + + p i0j 0 y j0 r + +p i0ny n r q i01y 1 r q i0j 1 y j1 r q i0ny n r 23 We have the following two cases: Case1 j 0 j 1 In this case, for the real numbers a i, 1 i n, c j0 and d j0 we define { yi r y i r a i, 1 i n, i j 0 y j0 r a j0 r, y j0 r c j0 r + d j0 24 in which a j0 > 0, c j0 < 0 and c j0 a j0 Then from 34 we will have n x i0 r P i0j q i0ja j + p i0j 0 a j0 r q i0j 0 c j0 r q i0j 0 d j0 j1 j j0

10 24 M S Hashemi, M K Mirnia and S Shahmorad x i0 r p i0j 0 a j0 q i0j 0 c j0 r + n j1 j j0 P i0j q i0ja j q i0j 0 d j0 Now it is sufficient to define the coefficient of the parameter r so that the function x i0 r is a decreasing function ie dxi 0 r dr < 0 Hence the coefficients c j0 and a j0 are determined so that p i0j 0 a j0 q i0j 0 c j0 < 0 Also, to preserve the fuzziness of Y, it is sufficient to have y j0 y j0, or, equivalently, a j0 r c j0 r + d j0 ; ie a j0 c j0 r d j0 or r dj 0 a j0 c j0 If we set d j0 a j0 c j0, then 0 r 1 Case2 j 0 j 1 In this case, for the real numbers a i, 1 i n, c j0 and d j0 we define y i r y i r a i, 1 i n, i j 0 y j0 r a j0 r, y j0 r c j0 r + d j0 y j1 r a j1 r, y j1 r c j1 r + d j1 25 where a j0 > 0, c j0 < 0, c j0 a j0, a j1 > 0, c j1 < 0 and c j1 a j1 From 34, we have n x i0 r P i0j q i0ja j + p i0j 0 a j0 r q i0j 0 c j0 r q i0j 0 d j0 j1 j j0,j 1 +p i0j 1 a j1 r q i0j 1 c j1 r q i0j 1 d j1 or x i0 r p i0j 0 a j0 + p i0j 1 a j1 q i0j 0 c j0 q i0j 1 c j1 r n + P i0j q i0ja j q i0j 0 d j0 q i0j 1 d j1 j1 j j0,j 1 Choosing a j0, a j1, c j0 and c j1 to satisfy p i0j 0 a j0 + p i0j 1 a j1 < q i0j 0 c j0 + q i0j 1 c j1 guarantees dxi 0 r dr < 0 Thus if d j0 a j0 c j0, d j1 a j1 c j1, the vector Y will be fuzzy, but the number x i0 r will not be fuzzy 4 Numerical Results Example 1: Consider the 2 2 fuzzy linear system { 2x 1 x 2 r, 3 r 3x 1 + 4x 2 r 1, 5 r

11 where A where Solving Fuzzy Linear Systems by Using the Schur Complement From 22 we have is obviously an M-matrix The extended 4 4 matrix is Z B ΞZ, C Clearly ΞZ, is an M-matrix, thus Z is an H-matrix By 33 1 P S 1 B CB 1 C Q P CB and thus Let so we have Z 1 P Q Q P X Y x1 r x 2 r r r 1 x1 r, X x 2 r 3 r, Y 5 r and from 33, 08 0 r 0 02 X P Y QY 0 04 r r r r 0 02 X P Y QY r r r + 2, 3 r 5 r r r 1

12 26 M S Hashemi, M K Mirnia and S Shahmorad and thus x 1 r x 1 r, x 1 r 06r + 1, 06r + 22, x 2 r x 2 r, x 2 r 02r + 14, 02r + 2 Obviously, x 1, x 2 are not fuzzy numbers Therefore the corresponding fuzzy solution is a weak fuzzy solution given by { u1 r 06r + 1, 06r + 22 u 2 r 12, 22 Example 2: Now consider a system where the coefficient matrix is the same as in the previous example, but where the fuzzy vector on the right hand side leads to a strongly fuzzy solution { 2x 1 x 2 2r, 3r + 5 where A x 1 + 4x 2 5r + 1, 4r + 10 and Z From the previous example we have P, Q, Z Also 2r 3r + 5 Y, Y 5r + 1 4r + 10 and thus from 39 it follows that r 0 02 X P Y QY r r r r X P Y QY r r r + 4 so that x 1 r x 1 r, x 1 r 08r + 2, 14r + 42, 3r + 5 4r r 5r + 1

13 Solving Fuzzy Linear Systems by Using the Schur Complement 27 x 2 r x 2 r, x 2 r 02r + 34, 04r + 4 Obviously x 1 r, x 2 r are fuzzy numbers for 0 r 1, and 2 x 1 r 28, 28 x 1 r 42, 34 x 2 r 36, 36 x 2 r 4 Therefore x , x , x , x x 1 r min{x 1 r, x 1 r, x 1 1, x 1 1}, x 1 r max{x 1 r, x 1 r, x 1 1, x 1 1}, x 2 r min{x 2 r, x 2 r, x 2 1, x 2 1}, x 2 r max{x 2 r, x 2 r, x 2 1, x 2 1}, so X is a strong fuzzy solution Example 3: Consider the 3 3 fuzzy linear system 6x 1 4x 2 x 3 r, 1 r x 1 + 5x 2 2x 3 r + 1, 3 r x x 3 r 2, r The coefficient matrix is which is an M-matrix Also, Y A r r + 1 r , Y The extended 6 6 crisp matrix is as follows: Z where and B ΞZ, C 1 r 3 r r

14 28 M S Hashemi, M K Mirnia and S Shahmorad Since ΞZ is an M-matrix, Z is an H-matrix Moreover, thus Z 1 P B CB 1 C 1 Q P CB and from 33 we have: ie X P Y QY X P Y QY,, r r r r r r ; x 1 r x 1 r, x 1 r r , r x 2 r x 2 r, x 2 r r , r x 3 r x 3 r, x 3 r r 4161, r which x 1, x 2, x 3 are fuzzy numbers, but u 1 r x 1 r, u 1 r x 1 r, thus the corresponding fuzzy solution is a weak fuzzy solution given by u 1 r u 1 r, u 1 r x 1 1, x , 4013 u 2 r u 2 r, u 2 r x 2 r, x 2 r r , r u 3 r u 3 r, u 3 r x 3 r, x 3 r r 4161, r + 844,

15 Solving Fuzzy Linear Systems by Using the Schur Complement 29 The authors thank the referees for their valuable sugges- Acknowledgement tions References 1 T Allahviranloo, The adomain decomposition method for fuzzy system of linear equation, Applied Mathematics and Computation, , O Axelson, Iterative solution methods, Cambridge University Press, J J Buckley and Y Qu, Solving systems of linear fuzzy equations, Fuzzy Sets and Systems , B N Datta, Numerical linear algebra and applications, Brooks/Cole Publishing Company, M Dehgan and B Hashemi, Iterative solution of fuzzy linear systems, Applied Mathematics and Computation, , D Dubois and H Prade, Fuzzy Sets and systems: Theory and Applications, Academic Press, New York, M Friedman, M Ming and A Kandel, Fuzzy linear systems, Fuzzy Sets and Systems, , C R Johnson, Inverse M-matrices, Linear Algebra and its Application, , M Ma, M Friedman and A Kandel, Duality in fuzzy linear systems, Fuzzy Sets and Systems, , M Otadi, SAbbasbandy and A Jafarian, Minimal solution of a fuzzy linear system, 6th Iranian Conference on Fuzzy Systems and 1st Islamic World Conference on Fuzzy Systemms, 2006, pp X Wang, Z Zhong and M Ha, Iteration algorithms for solving a system of fuzzy linear equations, Fuzzy Sets and Systems, , B Zheng and K Wang, General fuzzy linear systems, Applied Mathematics and Computation, in press M S Hashemi, M K Mirnia and S Shahmorad, Department of Applied Mathematics, Faculty of Mathematical Science, University of Tabriz, Tabriz-Iran address: hashemi math396@yahoocom address: mirnia-kam@tabrizuacir address: shahmorad@tabrizuacir Corresponding author