Abstract
We consider a class of fuzzy linear system of equations and demonstrate some of the existing challenges. Furthermore, we explain the efficiency of this model when the coefficient matrix is an -matrix. Numerical experiments are illustrated to show the applicability of the theoretical analysis.
1. Introduction
In the field of scientific and technical computation, various equations which describe realistic problems like natural phenomena or engineering problems such as computational fluid dynamics, finite differences methods, finite element methods, statistics, time/frequency domain circuit simulation, dynamic and static modeling of chemical processes, cryptography, magnetohydrodynamics, electrical power systems, differential equations, quantum mechanics, structural mechanics (buildings, ships, aircraft, and human body parts…), heat transfer, MRI reconstructions, vibroacoustics, linear and nonlinear optimization, financial portfolios, semiconductor process simulation, economic modeling, oil reservoir modeling, astrophysics, crack propagation, Google page rank, Gene page rank, 3D computer vision, cell phone tower placement, tomography, model reduction, nanotechnology, acoustic radiation, density functional theory, quadratic assignment, elastic properties of crystals, natural language processing, DNA electrophoresis, and so forth must be solved numerically. These problems can lead to solving a system of linear equations. There are many methods for solving linear systems; see [1–7] and the references therein.
Nevertheless, when coefficients of a system are ambiguous and there is some inexplicit information about the exact amount of parameters, one can solve a linear equation system by fuzzy logic. In 1965 [8], fuzzy logic was proposed by Zadeh and, following his work, many papers and books were published in fuzzy system theory. In particular, the solutions of fuzzy linear systems have been considered by many researchers, for example, [8–14]. Some investigations on the numerical solution of the fuzzy linear systems have also been reported; see [15–23] and references therein. Most of these studies use the same expansion of [12], where the coefficient matrix is crisp and the right-hand side is an arbitrary fuzzy number vector. Friedman et al. [12], using the embedding method, presented a general model for solving an fuzzy linear system and replaced the fuzzy linear system by crisp linear system. They studied also the uniqueness of the fuzzy solution for this model [12].
Dehghan and Hashemi [16] investigated the existence of a solution for this model under the condition that the coefficient matrix is a strictly diagonally dominant matrix with positive diagonal entries. Hashemi et al. [17], using the Schur complement, studied existence of the solution to this model when the coefficient matrix is -matrix and all diagonal entries are positive.
In this paper we establish the existence of the solution to Friedman et al.’s model under general class of the coefficient matrix called -matrix. It is well known that every strictly diagonally dominant matrix is an -matrix and every -matrix is an -matrix [2]. We demonstrate effectiveness of some conditions in our theorems (Section 3) relative to fuzzy linear systems.
2. Preliminaries
2.1. Fuzzy Numbers
An arbitrary fuzzy number is represented, in parametric form, by an ordered pair of functions , , which satisfy the following conditions (see [24]):(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), By appropriate definitions the fuzzy number space becomes a convex cone which is then embedded isomorphically and isometrically into a Banach space [12]. An alternative definition which yields the same is given by [10].
2.2. Fuzzy Linear System (FLS)
Definition 1. Consider the linear system of equations: where the coefficient matrix , , is a crisp matrix and , , is called a fuzzy linear system (FLS).
Definition 2. A fuzzy number vector , given by parametric form , , , is called a solution of the fuzzy linear system (2) if In order to solve the system given by (3), Friedman et al. [12] have solved a crisp linear system as where are determined as follows: and any which is not determined by (5) is zero. Then referring to Friedman et al. [12], we have or where , .
Theorem 3 (see [12]). Matrix is nonsingular if and only if matrices and are both nonsingular.
Theorem 4 (see [12]). Let be nonsingular. Then the unique solution is always a fuzzy vector for arbitrary vector if and only if is nonnegative.
Beside Theorem 3, the following theorem is also appropriate for proving the nonsingularity of matrix .
Theorem 5. Let be nonsingular. Then matrix in (6) is nonsingular if and only if Schur complement of , that is, , is nonsingular.
Proof. After some manipulations, the proof is obtained.
2.3. -Matrices and Their Subclasses
Let be an matrix, ; and for each nonempty subset of indices , we denote
Definition 6. For any matrix the comparison matrix is defined by
A real square matrix whose off-diagonal elements are all nonpositive is called an -matrix. Let be an -matrix; if then is said to be an -matrix.
A complex matrix is an -matrix if and only if is an -matrix.
Definition 7. Matrix is called an SDDM (strictly diagonally dominant matrix) if
Definition 8. Matrix is called a GDDM (generalized diagonally dominant matrix) if there exists a positive diagonal matrix such that is an SDDM matrix [3].
Theorem 9 (see [3]). Matrix is a GDDM if and only if is an -matrix.
Definition 10 (see [25]). Matrix is called a DDDM (doubly diagonally dominant matrix) if
Definition 11 (see [25]). For a given nonempty proper subset of indices matrix is called -SDDM (-strictly diagonally dominant matrix) if
Definition 12 (see [26]). -matrix is a Stieltjes matrix if is SPDM (symmetric and positive definite matrix).
The following are well known classes of nonsingular matrices, introduced by Ostrowski.
Definition 13. Matrix is called an matrix if there exists , such that, for each , it holds that .
Definition 14. Matrix is called an matrix if there exists , such that, for each , it holds that .
Theorem 15 (see [27]). If a matrix is or matrix, then it is nonsingular. Moreover it is an -matrix.
3. Some Theorems
In [12] the following points are dominant.(1)“ may be singular even if the original matrix is not.”(2)“The solution vector is unique but may still not be an appropriate fuzzy vector.”In this part we provide some sufficient conditions to avoid the above problems. We let the diagonal elements of be all positive.
In Figure 1, we describe the relations between -matrices and some of their subclasses. For more details please see [1, 25–27]. These relations are very important for our discussions and in Section 2.3 all items are completely defined.
The following theorems present the conditions that matrix needs to be an -matrix.
Theorem 16. Matrices in (6) and (7) are -matrixif and only if in (2) is an -matrix.
Proof. Let be an -matrix; then by Theorem 9, 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 SDDM. Therefore, by Theorem 9, is an -matrix too. Conversely, if is an -matrix, then by reasoning similar to that above, one can see that is an -matrix too.
Theorem 17. Matrices in (6) and (7) are -matrix if and only if in (2) is subclasses of -matrix.
Proof. By Section 2.3, Figure 1, and Theorem 16 proof is completed.
Theorem 18. For any , , for which there exists an index such that It follows that in (6) and (7) is an -matrix.
Proof. Based on Theorem 16 and applying Dashnic-Zusmanovich’s result [27, Theorem 5], the proof is completed.
Corollary 19. The unique solution is always a fuzzy vector for arbitrary vector , if is -matrix or subclasses of it.
Proof. By Definition 6, Theorem 4, Figure 1, and Theorems 1617 proof is completed.
4. Numerical Example
In this section, we give some examples of FLS to illustrate the results obtained in the previous sections.
Example 1. Consider the fuzzy system Since is an -matrix, by Theorem 17, is -matrix too. Therefore, by Theorem 4 and Corollary 19, we can solve the above FLS by Friedman et al.’s model. Now we solve this FLS: Therefore The exact solution is
Example 2. Consider the fuzzy system The extended matrix is where Evidently, is an -matrix and then, by Theorem 16, is also an -matrix. Therefore, we can solve this FLS by Friedman et al.’s model.
5. Conclusion
In this paper, we have studied a class of fuzzy linear system of equations, called Friedman et al.’s model. Furthermore, we proposed some theorems and effectiveness of some conditions in our theorems relative to fuzzy linear systems. Finally, from numerical experiment, we can see that our theorems are applicable and true.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank Dr. Anh-Huy Phan, editor of this journal, for the help and cooperation and also to thank all the three reviewers for their valuable suggestions which have led to an improvement in both the quality and clarity of this paper.