An Iterative Method for Solving a Symmetric System of Fuzzy Linear Equations

There are several iterative methods to solve fuzzy linear system of equations while the number of proposed methods for solving a crisp linear system of equations is much more. Surely, all of crisp methods will not be applicable for fuzzy problems, however, scientists attempt to develop the number of fuzzy solvers. In this study, one of these techniques based on chebyshev iteration is applied for solving fuzzy symmetric linear systems (FSLS). At the end of this paper, the new algorithm is illustrated by solving some numerical examples.

There are many applications for the solution of conclusion are drawn in the next sections. linear system of equations. Due to the importance of these solutions for various problems, scientists have Basic Concepts and Definitions: Here, some primary paid especial attention to solve such problems by definitions and notes, which are required in this study, iterative methods as ease and fast as possible. So, it have been indicated from [4][5][6][7]. is important to study more about variety of efficient techniques for solving linear system of equations [1][2][3].
Definition 1: The r-level set of a fuzzy set is defined as In some cases, at least some of the system's parameters an ordinary set [ ] of which the degree of membership are represented by fuzzy rather than crisp function exceeds the level r, i.e. numbers to fit more the problem with its nature that this problem is extracted from. The solution of such a (1) fuzzy linear system is applicable in different areas such as economic, engineering, physics etc. Therefore, it Definition 2: A fuzzy set , defined on the universal is essential to develop solving methods, which set of real number R, is said to be a fuzzy number if appropriately treat fuzzy linear systems, for finding its membership function has the following suitable solutions. characteristics: For the first time, the concept of fuzzy numbers and associated arithmetic operations was proposed by Zadeh.
is convex i.e. Next, a general model for solving a n×n fuzzy linear system of equations(FLSE), where the coefficient matrix is crisp and the right hand side is a fuzzy vector, was (2) proposed by Friedman et al [4,5]. After that, in literature, various methods have been proposed for solving FLSE is normal i.e. x R such that µ (x ). = 1 [6][7][8][9] in which many of them based on some popular solver µ is piecewise continuous. methods for crisp linear systems [10].
This paper is organized as follows. In section 2, we Definition 3: A fuzzy number in parametric form is a pair discuss some basic definitions and results on fuzzy of functions , , 0 r 1, that satisfies the numbers and the fuzzy linear system of equations.
following requirement: algorithm is discussed in section 3. Numerical tests and   .

Definition 4:
The addition and scalar multiplication of fuzzy numbers are defined by the extension principle and can be equivalently represented as follows, see Friedman et al. [6,7]. For arbitrary , and k R, the addition and the scalar multiplication are defined as follows: , Remark 1: A crisp number is simply represented by .

Definition 5:
The triangular fuzzy number is a fuzzy set where the membership function is as (3) and its parametric form is (4)

Definition 6: A triangular fuzzy number is said to be non-negative fuzzy number if and only if
For solving a n×n fuzzy linear system (5) with a crisp square matrix A and a triangular fuzzy vector different iterative methods have been proposed.
The ith row of fuzzy linear system with the solution i = 1,...,n is as and each s which is not determined by (7) is zero, ij and .
The matrix S is determined as a symmetric block matrix where b = s and c = s .
An approximate solution of (5) that is often used is the least square solution of (6), defined as a vector X which minimizes the Euclidean norm of (Y-SX). Solving Fuzzy Linear Systems: In this section, chebyshev iterative method [3] is applied for solving fuzzy symmetric linear system of equations. To have a better view, the chebyshev algorithm is recalled below from [3] for solving a crisp linear system of equations SX = Y.
In this algorithm, and are the largest and max min smallest eigenvalues of S in point of absolute value, respectively. This algorithm may use for nonsymmetric square matrices; however, the speed of convergence will be too slow. The approximated solution of this problem has been computed after 38 iterates.

CONCLUSION
Fuzzy linear system of equations is an applicable problem that its solution is important in many areas. I this paper, the chebyshev iteration method has been applied for finding the solution of fuzzy symmetric linear system of equations. Numerical example shows that this algorithm is practical for solving such problems.