Determination of the degrees of P-property and nonnegative invertibility for a fuzzy matrix

International Journal of Approximate Reasoning 46 (2007) 98–108 www.elsevier.com/locate/ijar

Determination of the degrees of P-property and nonnegative invertibility for a fuzzy matrix Mehdi Dehghan a

b

a,*

, Behnam Hashemi

a,b

Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran 15914, Iran Mathematics Department, Faculty of Basic Sciences, Shiraz University of Technology, Modarres Boulevared, Shiraz 71555-313, Iran

Received 25 April 2006; received in revised form 20 August 2006; accepted 7 September 2006 Available online 9 October 2006

Abstract Checking P-property and nonnegative invertibility of real and interval matrices has been widely investigated in the literature. In this paper, we try to extend the study to fuzzy matrices of fuzzy numbers, by employing the methods used for the corresponding interval matrices. To determine the degree of P-property and nonnegative invertibility for a fuzzy matrix, we propose an algorithm. Applications in matrix stability are given via the numerical examples.  2006 Elsevier Inc. All rights reserved. Keywords: Fuzzy matrix; Interval matrix; P-property; Nonnegative invertibility; Lyapunov diagonal stability

1. Introduction An n · n real (crisp) matrix A is called a P-matrix (or, to have the P-property) if all its principal minors are positive, where a principal minor of A is the determinant of a principal submatrix that is formed by arbitrary rows and the corresponding columns of A. For example let

*

Corresponding author. Tel.: +98 21 64542503; fax: +98 21 66497930. E-mail addresses: [email protected] (M. Dehghan), [email protected], [email protected] (B. Hashemi). 0888-613X/$ - see front matter  2006 Elsevier Inc. All rights reserved. doi:10.1016/j.ijar.2006.09.002

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2

2 A ¼ 4 3 4

99

3 0 1 1 0 5: 2 5

Considering det[2] = 2, det[1] = 1, det[5] = 5,     2 0 2 1 det ¼ 2; det ¼ 14; 3 1 4 5

 det

1

0

2 5

 ¼ 5;

det½A ¼ 8;

we see that all principal minors of A are positive. Therefore, A is a P-matrix. P-matrices play an important role in several areas, e.g., in the linear complementary theory since they guarantee existence and uniqueness of the solution of a linear complementarity problem (LCP). Given an n · n matrix M = (mij) and an n-vector q = (qi) the LCP, involves finding a vector z such that q þ Mz P 0;

z P 0;

T

ðq þ MzÞ z ¼ 0

or showing that no such vector z exists, where the matrix and vector inequalities are meant componentwise. The LCP has various applications, e.g., in circuit simulation, contact problems with friction, free boundary problems, linear and quadratic programming, finding a Nash-equilibrium in bimatrix games and optimal stopping in Markov chains [28]. In [28] an application of a linear complementarity problem with a P-matrix is presented. For a detailed introduction to the LCP we refer to [3]. P-matrices are also applied to localize the eigenvalues of a real matrix [23] or in global invertibility of vector fields [10]. In addition, P-matrices encompass four important subclasses of matrices. First, P-matrices contain totally positive matrices, where A is called totally positive if all the minors of A are positive. Second, P-matrices contain all positive definite matrices, where A = (aij) is called positive definite if xTAx > 0 for all 0 5 x 2 Rn. Third, they contain M-matrices, where A is called an M-matrix, if aij 6 0 for i 5 j and A1 P 0 [28]. Finally P-matrices contain the real diagonally dominant matricesPwith posii¼n tive diagonal entries, where A is called diagonally dominant, if jaij j P i¼1;i6¼j jaij j, j = 1, 2, . . . , n. For more information about these notions, the reader is invited to consult a textbook, for example [30]. The first systematic study of P-matrices appeared in the work of Fiedler and Ptak in 1966 [9]. Rohn and Rex have generalized the concept of P-property to interval matrices. They discussed the problem of checking P-property of interval matrices and necessary and sufficient conditions for an interval matrix to be a P-matrix [26]. Interval computations have various real-life applications from robotics, automatic control, image processing, astrophysics, traffic control and expert systems to ergonomics, social sciences and economics [15]. Further applications have also introduced in [14]. Rump designed the Interval Laboratory (INTLAB) which is a free useful software package for the interval computations. It can be found at http://www.ti3.tu-harburg.de/rump/intlab/. Another free interval arithmetic toolbox for Matlab is b4m (Basic interval arithmetic subroutines for MATLAB) that is available at http://www.ti3.tu-harburg.de/zemke/b4m/. An online website that listed all the interval softwares is http://www.cs.utep.edu/intervalcomp/intsoft.html. Checking properties of interval matrices has been widely investigated in the literature [16,18,20,24–27]. It is also known that checking regularity, positive definiteness, P-property and stability of interval matrices are NP-hard problems. In addition

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checking nonnegative invertibility and M-matrix property may be done in polynomial time [27]. Physical systems may be modeled by mathematical descriptions. In many practical situations, to measure the value of physical quantities we use regular measuring instruments. Measurements are commonly not absolutely accurate. So, instead of the actual value x of a physical quantity, we only know the measurement result x 0 . For a measuring instrument the manufacturer must provide it with some guaranteed error bounds d. Otherwise whatever for the value x 0 we measure, the actual value x can be arbitrarily far from x 0 . If we know the bound d, and the result of the measurement is x 0 , then the actual value must belong to the interval [x 0  d, x 0 + d]. On the other hand, in many practical situations, in addition to the intervals that are guaranteed by the manufacturer, it is reasonable to consider subintervals that the manufacturer cannot guarantee, but which the experts (designers or producers of the manufacturing instrument) claim to be true. To explain the practical usefulness of fuzzy sets, let us recall a real-life example where such subintervals are useful [22]. Earthquakes are extremely difficult to predict. Because of that, if we only use the equations and the measured data, we get very wide intervals of possible magnitudes and frequencies, intervals that are known to be much wider than the interval of actual values. So, if we use these wide intervals to design a building that is guaranteed to endure a typical earthquake, these buildings will be unnecessarily expensive. To make the requirements more realistic, in addition to the guaranteed measurement results, expert estimates are normally used that lead to narrower intervals. Also, note that different experts can have different ideas of which values are possible and which values are not [22]. Fuzzy methodology provides us with a way to represent expert estimates in a form understandable by a computer: namely, we represent an expert’s statement by a function (called membership function) that assigns to every real number x a degree of belief that x satisfies the expert estimate. Therefore, fuzzy numbers constitute a very interesting alternative to intervals that can represent those subintervals with their degrees of belief (the membership function, in fuzzy notion). A fuzzy number may emerge in a practical application as a description of the variation in our knowledge about the correct value of some measurements when the level of our confidence in that knowledge varies. Fuzzy systems including fuzzy set theory and fuzzy logic have many successful applications. Sophisticated fuzzy set theoretic methods have been applied to various areas ranging from fuzzy topological spaces to quantum optics, medicine and so on. On the other hand, in some engineering problems we need to check the stability of the corresponding system. Sometimes checking stability of these systems can be done via checking the P-property of a matrix. For instance, Lyapunov diagonal stability of a tridiagonal crisp matrix can be checked by its P-property. In many real-life cases, however, we do not know the exact values of the parameters that describe the system. Instead, we have expert estimates for these values that can be represented as fuzzy numbers. In these cases we need to know about the degree of P-property of the corresponding fuzzy matrix. Thus, it is important to develop procedures that would appropriately treat general fuzzy mathematical concepts. In this paper we try to extend the concept of P-property for fuzzy matrices. For finding the degree of P-property for fuzzy matrices, we utilize the works of Nguyen and Kreinovich [21], P-property criteria of interval matrices introduced by Rohn and Rex [26], and the method of Griffin and Tsatsomeros to compute all the principal minors of a real crisp

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matrix [11]. We establish an algorithm to determine the degree of P-property for fuzzy matrices. The bigger the degree of P-property, the greater is the possibility that the fuzzy matrix is a P-matrix. Section 2, reviews the basic points of the interval and fuzzy matrices. Section 3 outlines an algorithm of binary search type that can be used for determining the degree of P-property and the degree of nonnegative invertibility of fuzzy matrices. Section 4 applies the algorithm to numerical examples, including applications. Section 5 is devoted to a brief conclusion. 2. Notation and preliminary concepts The following notations will be used in this paper. a and A are shown a real (crisp) b are shown an interval number and an number and a real (crisp) matrix. Also, ^ a and A e are shown a fuzzy number and a fuzzy matrix, interval matrix. In addition e a and A respectively. b is a set of matrices whose elements range independently of each An interval matrix A other within prescribed bounds, i.e., the set of matrices of the form b ¼ ½A; A ¼ fA : A 6 A 6 Ag; A

ð1Þ

where the inequalities the center   are understood componentwise and A 6A. Introducing  matrix AC ¼ 12 A þ A and the nonnegative radius matrix D ¼ 12 A  A , we can also write the interval matrix (1) in the form b ¼ ½AC  D; AC þ D: A b is said to be symmetric if A b¼c An interval matrix A As where   1 1 c As ¼ ðA þ AT Þ; ðA þ AT Þ : 2 2 b ¼ ½A; A is symmetric if and only if the bounds A and A are It can be easily seen that A symmetric. Similarly, an interval matrix of the form [AC  D, AC + D] is symmetric if and only if both AC and D are symmetric. So, a symmetric interval matrix may contain nonsymmetric matrices. Let us introduce an auxiliary set Z ¼ fz 2 Rn jzj 2 f1; þ1g

for j ¼ 1; 2; . . . ; ng;

b ¼ ½A; A, Rohn defined matrices Az, i.e., the set of all ±1-vectors. For an interval matrix A z 2 Z by 1 1 ðAz Þij ¼ ðAij þ Aij Þ  ðAij  Aij Þzi zj 2 2

ði; j ¼ 1; 2; . . . ; nÞ:

b for each z 2 Z. Clearly, (Az)ij = Aij if zizj = 1 and ðAz Þij ¼ Aij if zizj = 1, hence Az 2 A Since Az = Az for each z 2 Z, the number of mutually different matrices Az is at most 2n  1 and equal to 2n  1 if A < A. Kreinovich [17] has noted that these special vertex matrices provide an optimal finite characterization of checking properties of interval matrices. b is called to be regular if each A 2 A b is nonsingular, and it is A square interval matrix A called singular if it contains a singular matrix. Regularity of interval matrices is an

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important property, because checking several properties of interval matrices can be reduced to checking regularity. A square matrix A is called stable if Re(k) < 0 for real part of the each eigenvalue k of b is called A. For symmetric matrices it is equivalent to kmax(A) < 0. An interval matrix A b stable (Hurwitz stable) if each A 2 A is stable. Stability of interval matrices has been widely studied in control theory due to its close connection to the problem of stability of the solution of a linear invariant system x_ ðtÞ ¼ AxðtÞ under data perturbations. For a survey of the necessary and sufficient conditions for stability of interval matrices, we refer the interested reader to the paper by Mansour [18]. Another property of a matrix is P-property that is important in the linear complemenb is called a P-matrix if each A 2 A b is a P-matrix. Rohn tary theory. An interval matrix A and Rex proved that a symmetric interval matrix is a P-matrix if and only if it is positive definite [26]. b is a P-matrix if and only if each Az, z 2 Z, is a Lemma 1 [26]. An interval matrix A P-matrix. Remark 2.1. Every n · n real (crisp) matrix has 2n  1 principal minors. Griffin and Tsatsomeros have developed, implemented and tested an algorithm (MAT2PM) to compute all the principal minors of a given n · n complex matrix [11]. In addition, we represent an arbitrary fuzzy number by a pair of functions ðuðrÞ; uðrÞÞ; 0 6 r 6 1 which satisfy the following requirements: • u(r) is a bounded left continuous nondecreasing function over [0, 1]. • uðrÞ is a bounded left continuous nonincreasing function over [0, 1]. • uðrÞ 6 uðrÞ; 0 6 r 6 1. A crisp number a is simply represented by uðrÞ ¼ uðrÞ ¼ a; 0 6 r 6 1. We can also show the triangular fuzzy numbers by triple (a, a, b), where a is the mean value of e a and a and b are left and right spreads, respectively. Thus u(r) will be the line between to points (a  a, 0) and (a, 1) in the (x, r) plane. Similarly, uðrÞ is the line between to points (a, 1) and (a + b, 0). In addition, a fuzzy number e a is called positive (negative), shown as e a > 0 (e a < 0), if its membership function l~a ðxÞ satisfies lea ðxÞ ¼ 0; 8x < 0ð8x > 0Þ: Thus e a ¼ ða; a; bÞ is positive, if and only if, a  a > 0. The fuzzy number e a ¼ ða; a; bÞ is shown in Fig. 1. The r-level set of a fuzzy set e a is defined as an ordinary set ½e a r of which the degree of membership function exceeds the level r: ½e a r ¼ fxjlea ðxÞ P r; r 2 ð0; 1g;

Fig. 1. A triangular fuzzy number.

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we separately define the 0-cut of e a , written ½e a 0 , as the closure of the union of the ½e a r , 0 < r 6 1. e ¼ ðf e is a fuzzy number [2,4– A matrix A aij Þ is called a fuzzy matrix if each element of A e will be positive (negative) and shown as A e > 0 (A e < 0) if each element of A e be 8,12]. A positive(negative). The nonnegative and nonpositive fuzzy matrices may be similarly defined. In this paper we use the triangular fuzzy numbers as the elements of fuzzy matrices. There is a strong connection between the fuzzy matrices of fuzzy numbers and the interval matrices, because each a-cut of a fuzzy matrix is an interval matrix (0 6 a 6 1). Interval matrices can in fact be considered as a special case of fuzzy matrices. Wang and Yan have extended the concept of Hurwitz stability to the fuzzy matrices [29]. Also, Dehghan et al. have introduced the concepts of a-regularity, (a, b)-regularity and regularity of a fuzzy matrix [6]. 3. The main part e denoted by p as 1  q where q Let us define the degree of P-property of a fuzzy matrix A e is a degree of belief that A is not a P-matrix. Hence in order to define p, we must define q. e is not a P-matrix if and only if one of the real matrices A = (aij) is not a A fuzzy matrix A e is not a P-matrix if and only if P-matrix. In other words A _ ða11 is a possible value of af 11 AND a12 is a possible value of af 12 AND .. . ann is a possible value of af nn AND A is not a P -matrixÞ; where _ means ‘‘or’’ extended to all combinatorics a11, a12 , . . . , ann of real (crisp) numbers. Let us use this formula to describe the degree of belief q that a matrix does not have the P-property. The degree of belief that aij is a possible value of f aij is described by the membership function l aeij ðaij Þ. Also, for given real numbers aij the P-property of a matrix A = (aij) is a crisp statement, thus its degree of belief is 1 (= true) or 0 (= false). Following standard fuzzy methodology, we interpret ‘‘AND’’ as min and ‘‘or’’ as max. Therefore we arrive at the following formula: q ¼ maxfmin½l ae ða11 Þ; l ae ða12 Þ; . . . ; l aenn ðann Þ; qða11 ; a12 ; . . . ; ann Þg; aij

11

12

where q(a11,a12, . . . , ann) is equal to 1 if the corresponding crisp matrix A = (aij) is not a P-matrix and to 0 if it has the P-property and max is taken over all possible matrices A. In addition, note that when q = 0, then min = 0, thus we do not need to consider these values when we compute max. Also, when q = 1, then since min(x, 1) = x, for every degree of belief x 2 [0,1], we do not need to consider this term in the min. Therefore, we can simplify the above formula into the following one: q ¼ maxfmin½l ae ða11 Þ; l ae ða12 Þ; . . . ; l aenn ðann Þg; aij

11

12

ð2Þ

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where max is taken over all aij for which this matrix is not a P-matrix. From (2), the degree of P-property of a fuzzy matrix can be defined by p = 1  q. Here we utter an algorithm to determine the degree of P-property of a fuzzy matrix e ¼ ðf A aij Þ. This algorithm consists of k iterations. On the jth iteration, we have an interval [h, h+] that contains q. After k iterations, we can take any endpoint of the resulting interval as the desired estimate for q, and compute p = 1  q. Algorithm 1 Step 1. Set h = 0, h+ = 1, N = 1, and e  1. Step 2. Compute h ¼ 12 ðhþ þ h Þ. Step 3. Find the real (crisp) values aij and aij for which l aeij ðaij Þ ¼ l aeij ðaij Þ ¼ h. So, the b ¼ ½aij ; aij  is formed, that corresponds to the h-cut of the fuzzy interval matrix A e matrix A. b Step 4. Use Lemma 1 to investigate the P-property of the interval matrix A. + b b Step 5. If A is an interval P-matrix, set h = h. Else if A is not an interval P-matrix, set h = h. Step 6. Take qN = h+ and compute pN = 1  qn. The accuracy is 2N. Step 7. If the accuracy 2N 6 e, then p = pN. Otherwise set N = N + 1 and return to Step 2. On each iteration, we detect the P-property of an interval matrix (Step 4); that by the use of Lemma 1, means to check the P-property of 2n  1 real crisp matrices. Totally we have k(2n  1) crisp matrices to check the P-property. Also, since each n · n crisp matrix Az has at most 2n  1 principal minors, we must check the positivity of at most k(2(n  1)(2n  1) real numbers. Therefore for example to determine the degree of belief p of a 4 · 4 fuzzy matrix with accuracy 0.01, we need to check the P-property of 56 crisp matrices, i.e., we need to determine and check the positivity of at most 840 real numbers as the principal minors. So, the most time-consuming part of this algorithm is checking weather an interval matrix has the P-property, i.e., Step 4. Also, the proof of the relation between (2) and Step 5 is mentioned below. Theorem 3.1. Let a nonnegative integer k be given. The above algorithm computes q with an accuracy 2k. Proof. By the use of the definition of q as the maximum, for each real number h, we have q > a, if and only if there exists the real numbers a11, a12, . . . , ann, such that min½l ae ða11 Þ; l ae ða12 Þ; . . . ; l aenn ðann Þ;  > h & A is not a P -matrix: 11

12

which is equal to the following term: l ae ða11 Þ > h& l ae ða12 Þ > h&    & l aenn ðann Þ > h & A is not a P -matrix: 11

12

Thus, q 6 a, if and only if for all real numbers a11, a12, . . . , ann, it is true that l ae ða11 Þ > h& l ae ða12 Þ > h&    & l aenn ðann Þ > h ! A is a P -matrix: 11 12

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This is what we check using the interval-case algorithm. The only difference is that the algorithm actually checks weather the following is true: l ae ða11 Þ P h& l ae ða12 Þ P h&    & l aenn ðann Þ P h ! A is a P -matrix: 11 12 The difference (P instead of >) can be taken care of, if we assume that the membership functions l aeij are continuous and monotonic (it decreases if we go further away from the mean value aij of the fuzzy number f aij ). h Now we want to apply our algorithm for computing the degree of nonnegative invertibility of a fuzzy matrix. A square matrix A is called nonnegative invertible if A1 P 0. An b is said to be nonnegative invertible if A1 P 0 for each A 2 A. b An interval interval matrix A b matrix A ¼ ½A; A is nonnegative invertible if and only if A and A are nonnegative invertible [24]. b ¼ ½A; A is nonnegative invertible if and only if A1 P 0 Lemma 2 [27]. An interval matrix A 1 and A P 0. Thus, we can use the above lemma to determine the degree of nonnegative invertibility for a fuzzy matrix of fuzzy numbers, in a similar way with the previous section. It can be done by using Algorithm 1 and the only difference in Steps 4 and 5, where we must use Lemma 2 instead of Lemma 1. 4. Applications and numerical examples One of the main problems to which the control theory is applied is stabilizing a plant. If we know exactly the parameters of the system that describe the plant dynamics, then we can use the standard methods of control theory to check whether the chosen control makes the plant stable or not. Lyapunov diagonally stable matrices play an important role in many disciplines, such as predator–prey system in ecology, economics and dynamic systems [13]. A 2 Rnn is called Lyapunov diagonally stable if there exists a positive definite diagonal matrix D such that AD + DAT is positive definite. Lemma 3 [1]. Let A 2 Rnn be an acyclic crisp matrix. Then A is Lyapunov diagonally stable if and only if A be a P-matrix. In many real-life cases, however, we do not know the exact values of the parameters that describe the plant dynamics. Instead, we have expert estimates for these values that can be represented as fuzzy numbers. For example, consider the following 3 · 3 fuzzy matrices of triangular fuzzy numbers: 2 3 ð5; 1; 1Þ ð4; 0; 1Þ ð0; 0; 0Þ e ¼ 4 ð3:5; 0:75; 0:7Þ A ð6; 1; 0:2Þ ð2:5; 0:5; 0:5Þ 5; ð0; 0; 0Þ ð5:5; 0:5; 0:25Þ ð7; 0:5; 1Þ 2 3 ð2; 1; 0:5Þ ð3; 0; 0:5Þ ð0; 0; 0Þ e ¼ 4 ð1:25; 0:35; 0:5Þ B ð3; 0:5; 0:3Þ ð1:5; 0:5; 0:6Þ 5: ð0; 0; 0Þ ð1:5; 0:25; 0:4Þ ð4; 0:5; 1Þ

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Fig. 2. The degree of P-property for two fuzzy matrices.

Since tridiagonal matrices are acyclic [1], using Lemma 3, checking the Lyapunov diage and B e is equivalent to checking onal stability of real crisp matrices in the fuzzy matrices A e and B e can be known as their P-property. So, the degree of P-property for fuzzy matrices A the degree of the Lyapunov diagonal stability of them. By using Algorithm 1, the degree of e and B e with N = 96 and accuracy 2 · 1030 is 0.3388 P-property of the fuzzy matrices A and 0.1767, respectively (see Fig. 2). So, we can conclude that the possibility of the e is greater than the possibility of the Lyapunov diagonal stability of the fuzzy matrix A e Lyapunov diagonal stability of the fuzzy matrix B. On the other hand, stable matrices are of great importance in the theory of differential equations. A crisp matrix is called stable if their eigenvalues all have negative real parts. An alternative criterion is the positivity of the Hurwitz determinants [19, pp. 185–186]. For 2 · 2 fuzzy matrices this criterion is equivalent to checking the P-property. As a test, consider the following fuzzy matrices: " #   ð3; 0; 1Þ ð1; 12 ; 1Þ ð3; 2; 1Þ ð1; 0; 1Þ e¼ e¼ A ; B : ð5; 13 ; 1Þ ð2; 1; 0Þ ð5; 2; 1Þ ð2; 1; 0Þ e and B e is 0.0774 and 0.1098, respectively. Therefore, The degree of P-property of A e is greater we can conclude that the possibility of the stability of the fuzzy matrix B e than the possibility of the stability of the fuzzy matrix A.

Fig. 3. Example 4.1 Left: The degree of P-property, Right: The degree of nonnegative invertibility.

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Example 4.1. Consider the following 2 · 2 fuzzy matrix of triangular fuzzy numbers:   ð3; 0; 0Þ ð1; 0; 3Þ e A¼ : ð3; 0; 0Þ ð2; 0; 0Þ e By using Algorithm 1, we conclude that the degree of P-property of the fuzzy matrix A with N = 96 and accuracy 2 · 1030 is 1.0000, while its degree of nonnegative invertibility is 0.3333. See Fig. 3. 5. Conclusion This paper tries to extend the concepts of P-property and nonnegative invertibility for fuzzy matrices of fuzzy numbers. Some numerical examples have also been given to show the implementation of the proposed algorithm. Acknowledgement The authors would like to offer particular thanks to one of the referees and Prof. Thierry Denoeux for their helpful comments and useful suggestions which have led to significant improvements of the paper. References [1] A. Berman, D. Hershkowitz, Graph theoretical methods in studying stability, AMS Summer Research Conference on Linear Algebra and its Role in System Theory, Bowdoin College, Maine, 1984. [2] J.J. Buckley, Y. Qu, Solving systems of linear fuzzy equations, Fuzzy Set. Syst. 43 (1991) 33–43. [3] R.W. Cottle, J.S. Pang, R.E. Stone, The Linear Complementarity Problem, Academic Press, New York, 1992. [4] M. Dehghan, B. Hashemi, M. Ghatee, Solution of the fully fuzzy linear systems using the decomposition procedure, Appl. Math. Comput., in press. [5] M. Dehghan, B. Hashemi, M. Ghatee, Solution of the fully fuzzy linear systems using iterative techniques, Chaos, Solitons & Fractals, in press. [6] M. Dehghan, M. Ghatee, B. Hashemi, Inverse of a fuzzy matrix of fuzzy numbers, submitted for publication. [7] D. Dubois, H. Prade, Systems of linear fuzzy constraints, Fuzzy Set. Syst. 3 (1980) 37–48. [8] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980. [9] M. Fiedler, V. Ptak, Some generalizations of positive definiteness and monotonicity, Numer. Math. 9 (1966) 163172. [10] D. Gale, H. Nikaido, The Jacobian matrix and global univalence of mappings, Math. Ann 19 (1965) 81–93. [11] K. Griffin, M. Tsatsomeros, Principal Minors, Part I: a method for computing all the principal minors of a matrix, Lin. Alg. Appl. 419 (2006) 107–124. [12] B. Hashemi, Topics in fuzzy numerical linear algebra. M.Sc. Thesis, Amirkabir University of Technology (Tehran Polytechnic), Tehran, April 2006. [13] D. Hershkowitz, Recent directions in matrix stability, Lin. Alg. Appl. 171 (1992) 161–186. [14] L. Jaulin, M. Kieffer, O. Didrit, E. Walter, Applied interval analysis, with examples in parameter and state estimation, Robust Control and Robotics, Springer-Verlag, London, 2001. [15] R.B. Kearfott, V. Kreinovich, Applications of Interval Computations, Kluwer Academic Publishers, Dordrecht, 1996. [16] V. Kreinovich, A. Lakeyev, J. Rohn, P. Kahl, Computational Complexity and Feasibility of Data Processing and Interval Computations, Kluwer, Dordrecht, 1997. [17] V. Kreinovich, Optimal finite characterization of linear problems with inexact data, Reliable Comput. 11 (2005) 479–489.

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