Iterative Methods for Solving Fuzzy Linear Systems

Australian Journal of Basic and Applied Sciences, 5(7): 1036-1049, 2011 ISSN 1991-8178

Iterative Methods for Solving Fuzzy Linear Systems 1,2

1

P. Mansouri, 1B. Asady

Department of Mathematics, Arak Branch, Islamic Azad University, Arak, Iran. 2 Department of Computer Science, Delhi University, New Delhi, India.

Abstract: Linear systems of equations, with uncertainty on the parameters, play a major role in several applications in various areas such as economics, finance, engineering and physics. In this paper, we proposed a method for solving a n×n fuzzy linear system to form AX =b. In which A is n×n nonsingular crisp matrix and b is arbitrary fuzzy vector. First we solved a fuzzy triangular system and by using of LU-Decomposition and/or Gaussian Elimination methods extend their methods for solving a general fuzzy linear system. Some examples are presented to illustrate these methods and compared with others methods. Key words: Fuzzy number, Fuzzy linear systems, Gaussian Elimination Method, LU-Decomposition, canonical form. INTRODUCTION Linear system of equations play a major role in many of areas of science such as engineering, mathematics, physics and economics. In many practical problems at least some of the system s parameters and measurements are fuzzy rather than crisp. Analyzing such systems requires the use of fuzzy information. Therefore, it is immensely important to develop mathematical models and numerical procedures that would appropriately treat fuzzy linear system and solve them. Linear system AX= b, where the elements aij of the matrix A and the elements bi of the vector b are fuzzy, is called a fuzzy linear system. Buckley (1991) proposed a method for solving it and S. Muzzioli and H. Reynaerts, (2006) generalized this method for solving the fuzzy linear system of the form A1 X  b1 = A2 X  b2 . Such R. Horcík, and A. Vroman and co worker, (2007) solved it with Cramer's rule. But, time of compute for solve a problem by Cramer's rule is excessive. A particular type of fuzzy system AX =b in which the coefficient matrix A is crisp and b a fuzzy vector investigated by many writers such as (Abbasbandy et al., 2006; Allahviranloo, 2005a; 2005b; 2003; Asady et al., 2005; 2009; Friedman, et al., 1998; 2000; Mizumoto, 1979; Nahmias, 1978; Nasseri and Ardil, 2005; Zheng and Wang, 2006) So that for solving it in (Abbasbandy et al., 2006; Allahviranloo, 2005a; 2005b; 2003; Asady et al., 2005; Friedman, et al., 1998; 2000; Zheng and Wang, 2006), they use embedding method given in (Cong-Xin and Ming, 1992) and replaced the original n×n fuzzy linear system by a (2n)×(2n) crisp function linear system. But unfortunately, sometimes their methods are unable to solve fuzzy linear system, since the (2n)×(2n) coefficient matrix may be singular even if the coefficient matrix A be nonsingular (see following example ). Example 1.: Consider the 2×2 fuzzy linear system as follows

 x1  x2 = y1 ,   x1  x2 = y2 . So that the matrix A is nonsingular, while the matrix 2×2

Corresponding Author: B. Asady, Department of Mathematics, Arak Branch, Islamic Azad University, Arak, Iran. E-mail: [email protected], [email protected].

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Aust. J. Basic & Appl. Sci., 5(7): 1036-1049, 2011

1  1 S = 0  0

0 0 1 1 0 0 1 1 0 0 1 1

  ,   

is singular. In others word, for some of linear equations, the proposed methods in (Abbasbandy et al., 2006; Allahviranloo, 2005a; 2005b; 2003; Asady et al., 2005; Friedman, et al., 1998; 2000) can not reach a solution or they offer an infinite numbers of solutions. In this paper we propose LU-Decomposition method for solving a (n×n) fuzzy linear system because triangular LU-Decomposition method is important for great practical to solving systems of linear equation. Therefore, This process would be developed through six sections. In section 2, we recall some fundamental results on fuzzy numbers and define fuzzy linear system. In section 3, we propose a method for solving a fuzzy linear triangular system and Gaussian elimination method and LU-Decomposition Method in section 4. Numerical examples are proposed in section 5 and last section is devoted to conclusions. 2 Preliminaries: Definition 1.: An arbitrary fuzzy number is represented by an ordered pair of functions u in parametric form is a pair

(u , u ) of function u (r ), u (r ), 0  r  1 , which satisfies the following requirements: • u (r )

is a bounded left continuous nondecreasing function over [0,1],

• u (r )

is a bounded right continuous nonincreasing function over [0,1],

• u (r )  u (r ), 0  r  1. The set of all these fuzzy numbers is denoted by E. Definition 2.: The n×n linear system

a11 x1  a x   21 1   a x   n1 1 

a12 x2 a22 x2  an 2 x2

   a1n xn = b1 ,    a2 n xn = b2 , (1)

   ann xn = bn ,

Or following matrix form Ax = b where the coefficient matrix A = (aij ),1  i  n

(2) and 1  j  n is crisp n×n matrix and bi  E , 1  i  n is

called a fuzzy system of linear equations(FLS). For arbitrary fuzzy numbers x = ( x ( r ), x ( r )), y = ( y ( r ), y ( r ))and real number k, we may define the addition and the scalar multiplication of fuzzy numbers by using the extension principle as

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x  y = ( x(r )  y (r ), x(r )  y (r )) k ( x, x) = (k x, k x), k  0,

(3)

k ( x, x) = (k x, k x), k < 0. 3 Solution of Fuzzy Linear Systems: Buckley and Qu, (1991) have been proposed a method to solve a fuzzy linear system Ax = b, where the elements, aij, of the matrix A and the elements, bi, of vector b are fuzzy numbers. They define the solution by, for all r0 [0,1]

(r ) = {t | (aij )t = y, aij  [a ij (r ), a ij (r )], yi  [bi (r ), bi (r )]}, and for all t  R n

xB (t ) = sup r | r  [0,1], t  (r ) . Clearly, if A was be a crisp matrix then we have

(r ) = {t | At = y, yi  [bi (r ), bi ( r )]}, for 0  r  1. We see that xB is defined as a fuzzy set on Rn and not as a vector of fuzzy numbers see. (Horcík) Therefore, xB expresses to what extent the real vector x is a solution of the system of linear equation, Ax = b. We prefer to define a solution as a vector of fuzzy numbers. In order to let us introduce our solution by parametric form to every component of the solution vector i,e.

( x)iB (r ) = ( xiB (r ), xiB (r )) where

xiB (r ) = min{xi |  x (r ) and xi is component of vector x},

(5)

xiB (r ) = max{xi |  x (r ) and xi is component of vector x},

(6)

for all i  {1, 2,..., n} . Now we have been proposed a method for solving of fuzzy triangular linear system (FTLS) to form Ux = b

(7)

in which U is a n×n crisp triangular matrix such that components of diagonal are nonzero (uii  0, i = 1, 2,..., n) and b an arbitrary fuzzy vector. Lemma 1: If U in (FTLS) (7) is a upper triangular matrix then, parametric form of the components solution is same as following

x n = min{b n /unn , b n /unn } , x n = max{b n /unn , b n /unn }, x k = min1/ukk {b k  (  ukj x j  ukj >0

x k = max1/ukk {b k  (  ukj x j  ukj >0

u

kj

ukj 0

1038

u

kj

x j )},

u

kj

x j )}.

ukj 0

kj

ukj 0

u

kj

x j )},

u

kj

x j )}.

ukj 0

kj

ukj 0

u

kj

x j )},

u

kj

x j )}.

ukj 0

u

kj

ukj 0

u

kj

x j )}

ukj 0

ukj 1) then rs1 = 0 enddo

(c1 , c1 ) = (b1 , b1 ) for j = 1: n  1do

rij = a1 j , (c1 , c1 ) = (b1 , b1 ) for i = j  1: ndo for k = j : ndo

uik = aik  (aij /a jj )a jk enddo if

(aij /a jj ) > 0

then

c i = min{bi  (aij /a jj )b j , bi  (aij /a jj )b j },

c i = max{bi  (aij /a jj )b j , bi  (aij /a jj )b j } eise

c i = min{bi  (aij /a jj )b j , bi  (aij /a jj )b j },

ci = max{bi  (aij /a jj )b j , bi  (aij /a jj )b j } enddo end do print(U , c ) enddo Recurrence algorithm for solving any triangular fuzzy linear system,with nonzero diagonal of coefficient triangular matrix is given by the following. Algorithm 2. Recurrence Algorithm for Solving Triangular Fuzzy Systems: Input:

A  C nn is nonsingular matrix,b is n×1 fuzzy vector is given to system (1)..

Output: × is n×1 fuzzy solution that satisfy to fuzzy linear system Ax = b.

LU  Decomposition( A, method = GaussianElimination) z1 , z1 ) = (b1 , b1 ) for i = 2: n do SUM 1 = 0; SUM 2 = 0

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for j =1; i-1 do

f (lij < 0)then

SUM 1 = SUM 1  lij z j

SUM 2 = SUM 2  lij z j else

SUM 1 = SUM 1  lij z j

SUM 2 = SUM 2  lij z j enddo

( z i , z i ) = (bi  SUM 1, bi  SUM 2) print zi = ( z i , z i ) enddo if (unn >0) then ( x n , x n ) = 1/unn ( z n , z n ) else ( x n , x n ) = 1/unn ( z n , z n ) for i = n-1 : 1 do SUM 3 = 0, SUM 4 = 0

for j = i+1 : n do if ((uij > 0) then SUM 3 = SUM 3  uij x j

SUM 4 = SUM 4  uij x j else SUM 3 = SUM 3  uij x j SUM 4 = SUM 4  uij x j

enddo

( x i , x i ) = 1/uii ({z i  SUM 3},{z i  SUM 4}) print

xi = ( xi , x i )

enddo REFERENCES Abbasbandy, S., R. Ezzati, A. Jafarian, 2006.LU decomposition method for solving fuzzy system of linear equations,Appl Math Comput., 172: 633-643. Allahviranloo, T., 2005. Solution of a fuzzy system of linear equation, Appl Math Comput., 155: 493-502. Allahviranloo, T., 2005. Successive over relaxation iterative method for fuzzy system of linear equations, Appl Math Comput., 62: 189-196. Allahviranloo, T., 2003. A comment of fuzzy system of linear equations, Fuzzy Sets and Systems., 140: 554. Asady, B., S. Abbasbandy, M. Alavi, 2005. Fuzzy general linear systems, Appl Math Comput., 169: 34-40. Asady, B., P. Mansouri, 2009. Numerical Solution of Fuzzy Linear System, International Journal of Computer Mathematics., 86: 151-162 .

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