Numerical methods for fuzzy system of linear equations
Introduction
Systems of simulations linear equations play major role in various areas such as mathematics, physics, statistics, engineering and social sciences. Since in many applications at least some of the system's parameters and measurements are represented by fuzzy rather than crisp numbers, it is important to develop mathematical models and numerical procedures that would appropriately treat general fuzzy linear systems and solve them. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [6], [2] and … One of the major applications using fuzzy number arithmetic is treating linear systems their parameters are all or partially represented by fuzzy numbers [1], [4] and … A general model for solving an n×n FLSE which coefficient matrix is crisp and the right hand side is arbitrary fuzzy number vector were first proposed by Friedman et al. [3]. They used the embedding method and replaced the original n×n fuzzy linear system AX=Y by a 2n×2n crisp linear system with a matrix S which may be singular even if A be nonsingular. The structure of this paper is organized as follows.
In Section 2, we bring some basic definitions and results on fuzzy numbers and fuzzy system of linear equation and (FSLE). In Section 3, we apply the iterative Jacobi and Gauss Sidel methods for solving FSLE with convergence theorems. The proposed algorithms are illustrated by solving some examples in Section 4 and conclusions are drawn in Section 5.
Section snippets
Preliminaries
We represent an arbitrary fuzzy number by an ordered pair of functions , which satisfy the following requirements:
- 1.
is a bounded left continuous nondecreasing function over [0,1].
- 2.
is a bounded left continuous nonincreasing function over [0,1].
- 3.
, 0⩽r⩽1.
A crisp number α is simply represented by . By appropriate definitions the fuzzy numbers space becomes a convex cone E1 which is then embedded isomorphically and isometrically in to a
The Jacobi and Gauss Sidel iterative techniques
First we are going to proof the following theorems. Theorem 3.1 Let S be nonsingular. Then the unique solution X of Eq. (2.5) is always a fuzzy vector for arbitrary vector Y, if S−1 is nonnegative. Proof It is sufficient to show that Definition 2.2 is hold for X. It is clear that S−1 have the same structure like S, i.e.from X=S−1Y, we haveThusbecause (S−1)ij=tij⩾0 and . Since is monotonically decreasing and is monotonically increasing, Eq. (3.1) is
Examples
Example 4.1 Consider the 2 × 2 fuzzy system The extended 4 × 4 matrix is The exact solutions are The exact and approximated solutions are plotted and compared in Fig. 1. Example 4.2 Consider the 3 × 3 fuzzy system
Conclusion
In this work we apply Jacobi and Gauss Sidel iterative methods for approximate of the unique solution of FSLE, since solving FSLE as analytically on the hole is difficult. We have assumed that the proposed matrix S by Friedman et al. [3] be nonsingular and aii>0, then we have proved that the matrix A is strictly diagonally dominant iff S be strictly diagonally dominant. If the unique solution of SX=Y is a strong or weak fuzzy number, then the approximate solution of iterative method also would
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