# H-Matrices in Fuzzy Linear Systems

Authors: H. Saberi Najafi; S. A. Edalatpanah
Journal: International Journal of Computational Mathematics
 (2014)

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## 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 anH-matrix. Numerical experiments are illustrated to show the applicability of the theoretical analysis.


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## Body

## 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 n×n fuzzy linear system and replaced the fuzzy linear system by 2n×2n 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 M-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 calledH-matrix. It is well known that every strictly diagonally dominant matrix is an M-matrix and every M-matrix is an H-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(u_(r),u¯(r)), 0≤r≤1, which satisfy the following conditions (see [24]):(i)
u_(r) is a bounded monotonic increasing left continuous function over [0,1];(ii)
u¯(r) is a bounded monotonic decreasing left continuous function over [0,1];(iii)
u_(r)≤u¯(r), 0≤r≤1.A crisp number α can be simply expressed as u_(r)=u¯(r)=α, 0≤r≤1. The addition and scalar multiplication of fuzzy numbers x=(x_(r),x¯(r)) and y=(y_(r),y¯(r)) can be described as follows:(i)
x=y if and only if x_(r)=y_(r) and x¯(r)=y¯(r),(ii)
x+y=(x_(r)+y_(r),x¯(r)+y¯(r)),(iii)
∀K∈R,
(1)Kx=(Kx_,Kx¯),K≥0,Kx¯,Kx_,K<0.By appropriate definitions the fuzzy number space {(u_(r),u¯(r))} becomes a convex cone E1 which is then embedded isomorphically and isometrically into a Banach space [12]. An alternative definition which yields the same E1 is given by [10].


### 2.2. Fuzzy Linear System (FLS)

Definition 1.
Consider then×n linear system of equations:
(2)a11x1+a12x2+⋯+a1nxn=b1,a21x1+a22x2+⋯+a2nxn=b2,⋮an1x1+an2x2+⋯+annxn=bn,
where the coefficient matrix A=(aij), 1≤i,j≤n, is a crisp matrix and bi∈E1, 1≤i≤n, is called a fuzzy linear system (FLS).Definition 2.
A fuzzy number vectorX=(x1,x2,…,xn)T, given by parametric form xi=(xi_(r),xi¯(r)), 1≤i≤n, 0≤r≤1, is called a solution of the fuzzy linear system (2) if
(3)∑j=1naijxj_=∑j=1naijxj_=bi_,∑j=1naijxj¯=∑j=1naijxj¯=bi¯.
In order to solve the system given by (3), Friedman et al. [12] have solved a 2n×2n crisp linear system as
(4)SX=B,
where S=(sij) are determined as follows:
(5)aij≥0⟹sij=aij,si+n,j+n=aij,aij<0⟹si,j+n=-aij,si+n,j=-aij
and any (sij) which is not determined by (5) is zero. Then referring to Friedman et al. [12], we have
(6)S=s1s2s2s1,X=x_-x¯,B=b_-b¯⟹s1x_-s2x¯=b_s2x_-s1x¯=-b,¯
or
(7)S=s1-s2-s2s1,X=x_x¯,B=b_b¯⟹s1x_-s2x¯=b_-s2x_+s1x¯=b,¯
where s1,s2∈Rn×n, A=s1-s2.Theorem 3 (see [12]).
MatrixS is nonsingular if and only if matrices A and s1+s2 are both nonsingular.Theorem 4 (see [12]).
LetS be nonsingular. Then the unique solution X is always a fuzzy vector for arbitrary vector B if and only if S-1 is nonnegative.Beside Theorem3, the following theorem is also appropriate for proving the nonsingularity of matrix S.Theorem 5.
Lets1 be nonsingular. Then matrix S in (6) is nonsingular if and only if Schur complement of A, that is, (TA=s1-s2s1-1s2), is nonsingular.Proof.
After some manipulations, the proof is obtained.


### 2.3.H-Matrices and Their Subclasses

LetA be an n×n matrix, N={1,2,…,n}; and for each nonempty subset Q of indices N, we denote
(8)riQA=∑k∈Q∖{i}aik,riA=riNA,liQA=∑k∈Qk<iaik,liA=liNA.Definition 6.
For any matrixA the comparison matrix M(A)=(mij)∈Rn×n is defined by
(9)mii=aii,mij=-aij,i≠j,i,j∈N.
A real square matrix whose off-diagonal elements are all nonpositive is called an L-matrix. Let A be an L-matrix; if A-1≥0 then A is said to be an M-matrix.
A complex matrixA is an H-matrix if and only if M(A) is an M-matrix.Definition 7.
MatrixA is called an SDDM (strictly diagonally dominant matrix) if
(10)aii>ri(A),i=1:n.Definition 8.
MatrixA is called a GDDM (generalized diagonally dominant matrix) if there exists a positive diagonal matrix W such that AW is an SDDM matrix [3].Theorem 9 (see [3]).
MatrixA is aGDDM if and only if A is an H-matrix.Definition 10 (see [25]).
MatrixA is called a DDDM (doubly diagonally dominant matrix) if
(11)aiiajj>riArjA,i,j=1:n,i≠j.Definition 11 (see [25]).
For a given nonempty proper subsetQ of indices N matrix A is called Q-SDDM (Q-strictly diagonally dominant matrix) if
(12)aii>riQ(A),i∈Q,aii-riQAajj-rjQ-A>riQ-ArjQA,i∈Q,j∈Q-=N∖Q.Definition 12 (see [26]).
L-matrix A is aStieltjes matrix if A is SPDM (symmetric and positive definite matrix).The following are well known classes of nonsingular matrices, introduced by Ostrowski.Definition 13.
MatrixA is called an α1 matrix if there exists α∈[0,1], such that, for each i∈N, it holds that |aii|>αri(A)+(1-α)ri(AT).Definition 14.
MatrixA is called an α2 matrix if there exists α∈[0,1], such that, for each i∈N, it holds that |aii|>ri(A)α+ri(AT)1-α.Theorem 15 (see [27]).
If a matrixA is α1 or α2 matrix, then it is nonsingular. Moreover it is an H-matrix.


## 2.1. Fuzzy Numbers

An arbitrary fuzzy number is represented, in parametric form, by an ordered pair of functions(u_(r),u¯(r)), 0≤r≤1, which satisfy the following conditions (see [24]):(i)
u_(r) is a bounded monotonic increasing left continuous function over [0,1];(ii)
u¯(r) is a bounded monotonic decreasing left continuous function over [0,1];(iii)
u_(r)≤u¯(r), 0≤r≤1.A crisp number α can be simply expressed as u_(r)=u¯(r)=α, 0≤r≤1. The addition and scalar multiplication of fuzzy numbers x=(x_(r),x¯(r)) and y=(y_(r),y¯(r)) can be described as follows:(i)
x=y if and only if x_(r)=y_(r) and x¯(r)=y¯(r),(ii)
x+y=(x_(r)+y_(r),x¯(r)+y¯(r)),(iii)
∀K∈R,
(1)Kx=(Kx_,Kx¯),K≥0,Kx¯,Kx_,K<0.By appropriate definitions the fuzzy number space {(u_(r),u¯(r))} becomes a convex cone E1 which is then embedded isomorphically and isometrically into a Banach space [12]. An alternative definition which yields the same E1 is given by [10].


## 2.2. Fuzzy Linear System (FLS)

Definition 1.
Consider then×n linear system of equations:
(2)a11x1+a12x2+⋯+a1nxn=b1,a21x1+a22x2+⋯+a2nxn=b2,⋮an1x1+an2x2+⋯+annxn=bn,
where the coefficient matrix A=(aij), 1≤i,j≤n, is a crisp matrix and bi∈E1, 1≤i≤n, is called a fuzzy linear system (FLS).Definition 2.
A fuzzy number vectorX=(x1,x2,…,xn)T, given by parametric form xi=(xi_(r),xi¯(r)), 1≤i≤n, 0≤r≤1, is called a solution of the fuzzy linear system (2) if
(3)∑j=1naijxj_=∑j=1naijxj_=bi_,∑j=1naijxj¯=∑j=1naijxj¯=bi¯.
In order to solve the system given by (3), Friedman et al. [12] have solved a 2n×2n crisp linear system as
(4)SX=B,
where S=(sij) are determined as follows:
(5)aij≥0⟹sij=aij,si+n,j+n=aij,aij<0⟹si,j+n=-aij,si+n,j=-aij
and any (sij) which is not determined by (5) is zero. Then referring to Friedman et al. [12], we have
(6)S=s1s2s2s1,X=x_-x¯,B=b_-b¯⟹s1x_-s2x¯=b_s2x_-s1x¯=-b,¯
or
(7)S=s1-s2-s2s1,X=x_x¯,B=b_b¯⟹s1x_-s2x¯=b_-s2x_+s1x¯=b,¯
where s1,s2∈Rn×n, A=s1-s2.Theorem 3 (see [12]).
MatrixS is nonsingular if and only if matrices A and s1+s2 are both nonsingular.Theorem 4 (see [12]).
LetS be nonsingular. Then the unique solution X is always a fuzzy vector for arbitrary vector B if and only if S-1 is nonnegative.Beside Theorem3, the following theorem is also appropriate for proving the nonsingularity of matrix S.Theorem 5.
Lets1 be nonsingular. Then matrix S in (6) is nonsingular if and only if Schur complement of A, that is, (TA=s1-s2s1-1s2), is nonsingular.Proof.
After some manipulations, the proof is obtained.


## 2.3.H-Matrices and Their Subclasses

LetA be an n×n matrix, N={1,2,…,n}; and for each nonempty subset Q of indices N, we denote
(8)riQA=∑k∈Q∖{i}aik,riA=riNA,liQA=∑k∈Qk<iaik,liA=liNA.Definition 6.
For any matrixA the comparison matrix M(A)=(mij)∈Rn×n is defined by
(9)mii=aii,mij=-aij,i≠j,i,j∈N.
A real square matrix whose off-diagonal elements are all nonpositive is called an L-matrix. Let A be an L-matrix; if A-1≥0 then A is said to be an M-matrix.
A complex matrixA is an H-matrix if and only if M(A) is an M-matrix.Definition 7.
MatrixA is called an SDDM (strictly diagonally dominant matrix) if
(10)aii>ri(A),i=1:n.Definition 8.
MatrixA is called a GDDM (generalized diagonally dominant matrix) if there exists a positive diagonal matrix W such that AW is an SDDM matrix [3].Theorem 9 (see [3]).
MatrixA is aGDDM if and only if A is an H-matrix.Definition 10 (see [25]).
MatrixA is called a DDDM (doubly diagonally dominant matrix) if
(11)aiiajj>riArjA,i,j=1:n,i≠j.Definition 11 (see [25]).
For a given nonempty proper subsetQ of indices N matrix A is called Q-SDDM (Q-strictly diagonally dominant matrix) if
(12)aii>riQ(A),i∈Q,aii-riQAajj-rjQ-A>riQ-ArjQA,i∈Q,j∈Q-=N∖Q.Definition 12 (see [26]).
L-matrix A is aStieltjes matrix if A is SPDM (symmetric and positive definite matrix).The following are well known classes of nonsingular matrices, introduced by Ostrowski.Definition 13.
MatrixA is called an α1 matrix if there exists α∈[0,1], such that, for each i∈N, it holds that |aii|>αri(A)+(1-α)ri(AT).Definition 14.
MatrixA is called an α2 matrix if there exists α∈[0,1], such that, for each i∈N, it holds that |aii|>ri(A)α+ri(AT)1-α.Theorem 15 (see [27]).
If a matrixA is α1 or α2 matrix, then it is nonsingular. Moreover it is an H-matrix.


## 3. Some Theorems

In [12] the following points are dominant.(1)
“S may be singular even if the original matrix A 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 A be all positive.In Figure1, we describe the relations between H-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.Figure 1
Relations betweenH-matrices and some of their subclasses.The following theorems present the conditions that matrixS needs to be an H-matrix.Theorem 16.
MatricesS in (6) and (7) are H-matrixif and only if A in (2) is an H-matrix.Proof.
LetA be an H-matrix; then by Theorem 9, there exists a positive diagonal matrix W such that AW is strictly diagonally dominant matrix. Without loss of generality, let AW be row strictly diagonally dominant; that is,
(13)∑j=1j≠inaijwjj<aiiwii.
Now, let
(14)j+=aij∣aij≥0,j-=aij∣aij<0,1≤i,j≤n;
then we have
(15)∑j=1j≠inaijwjj=∑j∈j+j≠iaijwjj+∑j∈j-j≠iaijwjj<aiiwii.
By considering the structure of S and since, for all i=1,…,n, aii≥0, we have
(16)∑j=1j≠insij=∑j∈j+j≠iaij,∑j=n+12nsij=∑j∈j-aij,1≤i≤n,∑j=1nsij=∑j∈j-aij,∑j=n+1j≠i2nsij=∑j∈j+j≠iaij,n+1≤i≤2n.
Therefore,
(17)∑j=1j≠insijwjj=∑j∈j+j≠iaijwjj+∑j∈j-j≠iaijwjj<aiiwii1≤i≤n,∑j∈j-j≠iaijwjj+∑j∈j+j≠iaijwjj<aiiwiin+1≤i≤2n⟹∑j=1j≠i2nsijwjj<aiiwii=siiwii.
Then by choice of T=W00W2n×2n, ST is row SDDM. Therefore, by Theorem 9, S is an H-matrix too. Conversely, if S is an H-matrix, then by reasoning similar to that above, one can see that A is an H-matrix too.Theorem 17.
MatricesS in (6) and (7) are H-matrix if and only if A in (2) is subclasses of H-matrix.Proof.
By Section2.3, Figure 1, and Theorem 16 proof is completed.Theorem 18.
For anyA=[aij]∈Rn×n, n≥2, for which there exists an index i∈Nsuch that
(18)aii·ajj-rjA+aji>riA·aji,∀j≠i,j∈N.
It follows that S in (6) and (7) is an H-matrix.Proof.
Based on Theorem16 and applying Dashnic-Zusmanovich’s result [27, Theorem 5], the proof is completed.Corollary 19.
The unique solutionX=S-1B is always a fuzzy vector for arbitrary vector Y, if A is M-matrix or subclasses of it.Proof.
By Definition6, Theorem 4, Figure 1, and Theorems 16-17 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 the2×2 fuzzy system
(19)3x1-x2=(3r,5-2r),-x1+2x2=(1+r,7-5r).
Since A is an M-matrix, by Theorem 17, S is M-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:
(20)A=3-1-12⟶A-1=0.40.20.20.6→5S=300-102-100-130-1002⟹S-1=0.4000.200.60.2000.20.400.2000.6.
Therefore
(21)X=x_1(r)x_2(r)x¯1(r)x¯2(r)=S-1B=1.4+0.2r1.6+0.2r2.2-0.6r4.2-2.4r.
The exact solution is
(22)x1=(x_1(r),x¯1)=(1.4+0.2r,2.2-0.6r),x2=(x_2(r),x¯2)=(1.6+0.2r,4.2-2.4r).Example 2.
Consider then×n fuzzy system
(23)3x1-x3=(2+r,4-r),x2+3x3-x4=(2+r,4-r),x3+3x4-x5=(2+r,4-r),⋮xn-3+3xn-2=(2+r,4-r),xn-2+3xn-1=(2+r,4-r),xn-1+3xn-x1=(2+r,4-r).
The extended 2n×2n matrix is
(24)S=s1-s2-s2s1,
where
(25)s1=30⋯000013⋯0000⋮⋱⋱⋯⋱⋱⋮0⋯130⋯00⋯013000⋯⋱01300⋯⋯0013,s2=000-10⋯00000-1⋱⋮⋮⋮⋱⋱⋱⋱00⋯⋯00⋱-10⋯⋯00000⋯⋯0000-10⋯0000.
Evidently, A is an H-matrix and then, by Theorem 16, S is also an H-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.



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