Abstract
We consider a class of fuzzy linear systems (FLS) and demonstrate some of the existing methods using the embedding approach for calculating the solution. The main aim in this paper is to design a class of mixed type splitting iterative methods for solving FLS. Furthermore, convergence analysis of the method is proved. Numerical example is illustrated to show the applicability of the methods and to show the efficiency of proposed algorithm.
1. Introduction
Solving fuzzy systems has been considered by many researchers, for example, [1–9] and the references therein. In [1, 2] Kandel et al. applied the embedding method for fuzzy linear system (hereafter denoted by FLS) and replaced the FLS by a crisp linear system. This model has been modified later by some other researchers; see [10–16] and the references therein.
Here, based on mixed type splitting, we introduce a new iterative method to FLS. The mixed type splitting iterative method [17, 18] is given for the linear system of equations , where is positive real. Cheng et al. in [19] presented a class of the mixed type splitting iterative methods based on [17, 18] and some convergence conditions were given. They also proposed some sufficient and necessary conditions of convergence when coefficient matrix of the linear system is certain matrices such as -matrix.
In this paper, the mixed type splitting iterative method for FLS will be established, which is a generalization of mixed type splitting iterative method for linear system. Some sufficient conditions for convergence of the mixed type splitting iterative method will be considered. Moreover, we will discuss a comparison theorem, which describes the influences of the parameters on the convergence rates of the new methods.
2. Preliminaries
In this section we provide some basic notations and definitions of fuzzy number and fuzzy linear system.
Definition 1. An arbitrary fuzzy number is represented, in parametric form, by an ordered pair of functions , , which satisfy the following requirements (see [2, 3]). (i) is a bounded monotonic increasing left continuous function over .(ii) is a bounded monotonic decreasing left continuous function over .(iii), .
A crisp number can be simply expressed as , . The addition and scalar multiplication of fuzzy numbers and can be described as follows:(i) if and only if and ;(ii); (iii)for all ;
Definition 2. Consider the linear system of the following equations: where the coefficient matrix , , , is a crisp matrix and , , is called a FLS.
Definition 3. A fuzzy number vector , given by the parametric form , , , is called a solution of the FLS (1) if
Friedman et al. [2, 3], in order to solve the system given by (2), have solved a crisp linear system as follows:
where are determined as follows:
and any which is not determined by (4) is zero.
Then referring to [2, 3] we have
Or
where and .
Definition 4 (see [20]). (a) A matrix is called a -matrix if, for any , .
(b) A -matrix is an -matrix, if .
(c) A -matrix is an -matrix, if is nonsingular and .
(d) For any matrix the comparison matrix is defined by
(e) A complex matrix is an -matrix if is an -matrix.
(f) Matrix is called a Generalized Diagonally Dominant Matrix (GDDM) if there exists a positive diagonal matrix such that is a Strictly Diagonally Dominant Matrix ().
Lemma 5 (see [19]). Let be the coefficient matrix of the linear system . If is an -matrix and conditions of (19) are satisfied, then the mixed type splitting iterative method is convergent.
Remark 6. For any splitting, , where is nonsingular, the iterative method for solving linear systems of is This iterative process converges to the unique solution for initial vector value if and only if the spectral radius , where is called the iteration matrix [20]. For example, suppose and , where and are strictly lower and strictly upper triangular part of , respectively. Then for classical AOR (see; [10]) we have And when , we have SOR method [20]; that is,
Lemma 7 (see [19]). Let be the coefficient matrix of the linear system . If is an L-matrix, conditions of (19) are satisfied and Then (i)If .(ii)If .(iii)If .
Lemma 8 (see [20]). Matrix is a GDDM matrix if and only if is an -matrix.
3. The Mixed Type Splitting Iterative Methods for Fuzzy Linear Systems
Let be nonsingular and , where and , are strictly lower and upper triangular matrices of , respectively. The iterative method for is where , det , and is any initial vector. There are several well-known point iterative methods and block numerical iterative methods for FLS such as Jacobi, Gauss-Seidel, SOR, and AOR; see [9–14]. As a matter of fact, these methods are generalization of iterative methods for crisp linear systems . For instance, in AOR method for FLS [12] we have where the iterative matrix is Therefore, we obtain And we have Other methods are the same and we know that, by choosing special parameters, the similar results can be obtained, for example,(1)Jacobi method for , ;(2)JOR (Jacobi Overrelaxation) method for ;(3)Gauss-Seidel method for ;(4)SOR method for .
Now, from we have the following algorithms (mentioned in [12] for the first time).
Algorithm 9.
AOR iterative method for FLS.
Step 1. Choose an initial vector and parameters and .
Step 2. For do
Step 3. If or , then stop; otherwise set and go to Step 2.
Step 4. End for.
Step 5. Return to .
Next, we will establish the mixed type splitting iterative method for FLS. The mixed type splitting iterative methods [19] for solving are given by the following: whose iteration matrix is Now, we consider the mixed type splitting iterative methods for solving FLS. Based on above demonstration, we have where and . Now, let Therefore, we obtain And we have Now, from we have the following algorithms (mentioned in [12] for the first time).
Algorithm 10. Mixed type splitting iterative methods_1 for FLS.
Step 1. Choose an initial vector and parameters and .
Step 2. For do
Step 3. If and , then stop; otherwise set and go to Step 2.
Step 4. End for.
Step 5. Return to .
Note that when we have SOR method for FLS (see [11]) and when where are real parameters with , we have AOR method for FLS (see [12]).
Theorem 11. Let be an -matrix and let conditions of Lemma 7 be satisfied.
Then we have
Proof. We only prove (i); (ii) can be similarly verified.
Let be an -matrix. Then is an -matrix and by Lemmas 5 and 7,
By definition of mixed splitting method, we have
Therefore we have
And the proof is completed.
Lemma 12. The Matrix in (5) or (6) is an -matrix if and only if in (1) is -matrix.
Proof. Let be an -matrix; then by Lemma 8, there exists a positive diagonal matrix such that is strictly diagonally dominant matrix. Without loss of generality, let be row strictly diagonally dominant; that is, Now, let then we have By considering the structure of and since, for all , , we have Therefore Then, by choice of is row . Therefore, by Lemma 8, is also an -matrix. Conversely, if is an -matrix, then by reasoning similar to that above, it can be seen that is an -matrix too.
Therefore, we can obtain the following theorem.
Theorem 13. Let be an -matrix and the following conditions are satisfied: Then the speed of convergence of Algorithm 9 is faster than the speed of convergence of SOR and AOR methods for FLS.
Proof. Let be an -matrix; then by Lemma 12, is an -matrix too. Furthermore, by Theorem 11, . Moreover, similar to the proving process of Theorem 11 and (31), we can obtain Therefore, and by Lemma 7 the proof is completed.
In the sequel, we will improve Algorithm 10 by the following updating technique.(I)First, use and from our previous information to compute the new exactly as in the first line of Step 2.(II)Then, replace in the second line of Step 2 with , obtained just as in the above, to compute the new .
Algorithm 14. Mixed type splitting iterative methods_2 for FLS.
Step 1. Choose an initial vector and parameters and .
Step 2. For do
Step 3. If or , then stop; otherwise set and go to Step 2.
Step 4. End for.
Step 5. Return to .
4. Numerical Example
In this section, we give an example of FLS to illustrate the results obtained in previous sections.
Example 1. Consider the fuzzy system
The extended matrix is
where
Evidently, is an -matrix and therefore is also an -matrix. Table 1 shows the numerical results of the above example with the tolerance . Moreover, the initial approximation is zero vector. In the Table 1, we reported the number of iterations (Iter), Elapsed time (ELP), and associated spectral radii () for the iterative methods with different , , and . This results show that our algorithms can be applied to a large class of FLS. Furthermore, from Table 1, we can see that for solving FLS, Algorithm 14 is superior to the other methods from point of view rate of convergence.
This example is computed with MATLAB7 on a personal computer Pentium 4-256 MHZ.
5. Conclusion
In this paper, we have proposed a class of splitting methods, called mixed type splitting iterative method for fuzzy linear systems that contains auxiliary matrices. Furthermore, our method with some other iterative methods is compared in the frame of fuzzy linear systems and it is shown that by proper choice of these auxiliary matrices, the new iterative method is faster than these mentioned methods form point of view of the convergence speed.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.