A relaxed Kaczmarz method for fuzzy linear systems

A relaxed Kaczmarz method is presented for solving a class of fuzzy linear systems of equations with crisp coeﬃcient matrix and fuzzy right-hand side. The iterative scheme is established and the convergence theorem is provided. Numerical examples show that the method is eﬀective.


Introduction
Fuzzy linear systems (FLSs) occur in many fields, such as control problems, information, physics, statistics, engineering, economics, finance and even social sciences [12]. Thus, it is significant to study the numerical methods for solving FLSs.
The rest of the paper is organized as follows. Section 2 gives some basic definitions and results of FLS. In Section 3, the relaxed Kaczmarz method is established with convergence theorem. Two numerical examples in Section 4 are discussed and the conclusion is in Section 5.
Definition 2.1. [12] A fuzzy number vector X = (x 1 , x 2 , · · · , x n ) T given by is called a solution of the fuzzy linear system (1.1) if a ij x j = y i . where S = (s kl ), s kl are determined as follows a ij 0 ⇒ s ij = a ij , s n+i, n+j = a ij , a ij < 0 ⇒ s i, n+j = a ij , s n+i, j = a ij , 1 i, j n, and any s kl which is not determined by the above items is zero, 1 k, l 2n, and Further, the matrix S has the structure 2) can be rewritten as A theorem as follows indicates when FLS (1.1) has a unique solution.
Theorem 2.2. The matrix S is nonsingular if and only if the matrices A = S 1 + S 2 and S 1 − S 2 are both nonsingular. See [12].
In the next section, the proposed relaxed Kaczmarz method is presented for nonsingular FLS (1.1).

The relaxed Kaczmarz method
For nonsingular fuzzy linear system (2.2) or (2.3), a relaxed Kaczmarz iterative scheme can be described as follows, and can be implemented as the following algorithm, where 0 < α < 2 is a relaxation parameter.
The convergence result for method (3.2) is as the following theorem. , generated by the relaxed Kaczmarz method (3.2) starting from an initial guess X 0 X 0 with X 0 and X 0 in the column space of S 2 , converges to the unique solution Moreover, the solution error for the iteration sequence is where λ min (·) is the smallest nonzero eigenvalues of a matrix, X k = X k X k , and X * = X * X * .
, thus . As x 0 is in the column space of S, from [23], it holds that S (X k − X * ) . Therefore, the following is obtained The proof is completed.

Numerical Examples
This section gives two examples to show the effectiveness of the relaxed Kaczmarz method. All implements using Matlab 7 run in a Windows 7 DELL laptop with Intel 2.80GHz CPU and 8.00GB RAM. In the numerical experiments, the initial guess is 0 and the stopping criterion is where R k is the residual vector after k iterations, i.e., R k = Y − SX k . In the tables, x a and x b mean that SX = Y is solved as two numeric systems (1 + r, 1 + r) (2 + r, 2 + r) . . .
(n 2 + r, n 2 + r)   6 give the number of iterations (IT) and the residual of the stopping step (RES). As n increases, the method requires more iterations, and with different α the method has different convergence rate, thus, improvement should be made and the optimal parameter should be studied to change the convergence.

Conclusion
A relaxed Kaczmarz method is presented for solving n × n fuzzy linear system. The numerical results show that the method is effective. Further work would be improving the method and comparing with other methods, also exploring the optimal relaxation parameter.

Compliance with Ethical Standards
Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.