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international journal of innovative technology and exploring engineering ijitee issn 2278 3075 online volume 9 issue 2 december 2019 finding inverse of a fuzzy matrix using eigen value method hamed ...

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                                                                 International Journal of Innovative Technology and Exploring Engineering (IJITEE)
                                                                                                     ISSN: 2278-3075 (Online), Volume-9 Issue-2, December 2019
                          Finding Inverse of a Fuzzy Matrix using Eigen 
                                                                                 value Method 
                                                                   Hamed Farahani, M. J. Ebadi, Hossein Jafari  
                  
                    Abstract: The present paper extends a concept of the inverse of                           The study of the second class is ignored because of the 
                 a  matrix  that  its  elements  are  fuzzy  numbers,  which  may  be                      complex arithmetic structure. In the present paper, the focus 
                 implemented to model imprecise and uncertain features of the                              will be on this fuzzy matrices class. The investigations of 
                 problems in the real world. The problem of inverse calculation of                         the invertibility of the square interval matrices and obtaining 
                 a  fuzzy  matrix  is  converted  to  solving  a  fuzzy  polynomial                        their inverse are two popular problems that have been warm 
                 equations (FPEs) system. In this approach, the fuzzy system is                            issues in recent studies. This paper proposes an approach to 
                 transformed  to  an  equivalent  system  of  crisp  polynomial                            compute the fuzzy inverse matrix on the base of eigenvalue 
                 equations. The solutions of the crisp polynomial equations system                         approach. In this method, finding the fuzzy inverse matrix is 
                 is computed using eigenvalue method. Also, using Gröbner basis                            on  the  base  of  changing  the  fuzzy  matrix  into  a  crisp 
                 properties  a  criteria  for  invertibility  of  the  fuzzy  matrix  is 
                 introduced. Furthermore, a novel algorithm is proposed to find a                          polynomial  equations  system.  Then,  a    Gröbner  basis 
                 fuzzy  inverse  matrix.  Achieving  all  entries  of  a  fuzzy  inverse                   regarding to a term order may be computed for the ideal 
                 matrix  at  a  time  is  a  big  advantage  comparing  the  existence                     which is  generated  by  the  crisp  polynomials  system.  The 
                 methods. In the end, some illustrative examples are presented to                          Gröbner basis regarding each arbitrary term order can be 
                 demonstrate the algorithm and concepts.                                                   computed. Moreover, the  Gröbner basis computation w.r.t 
                    Keywords : Eigenvalue, Fuzzy numbers, Fuzzy matrix, Fuzzy                              the lexicographical term order regarding the other terms of 
                 identity matrix, Fuzzy linear equation system.                                            the order is more complex in the computational complexity 
                                           I.      INTRODUCTION                                            viewpoint [1]. Hence, a suitable term order can be selected 
                                                                                                           for  computing  the    Gröbner  basis  and  reducing  the 
                     When the fuzzy uncertainty happens in a problem, the                                  computations  rate.  In  the  eigenvalue  approach,  the  roots 
                 fuzzy  matrices  are  successfully  applied.  In  the  last  two                          calculation  of  a  system  is  accomplished  separately  from 
                 decades, fuzzy matrices have been popular [24]. In matrix                                 each other. Accordingly, the occurred approximation and the 
                 theory, the position of the theory of generalized inverse of a                            probable  error  in  the  previous  roots  calculation  don’t 
                 fuzzy  matrix  is  outstanding  [5,  6].  The  research  of                               influence  the  next  roots  computation.  In  the  presented 
                 convergence  of  powers  of  a  fuzzy  matrix  began  by                                  approach,  the  inverse  computation  of  a  fuzzy  matrix  is 
                 Thomasan [22] in 1977. A systematic improvement to the                                    converted  to  obtaining  the  eigenvalues  of  a  matrix. 
                 fuzzy  matrix  theory  was  given  by  Kim  and  Roush  [20].                             Therefore, the valuable tools from linear algebra can be used 
                 Also, they proposed the algorithms to obtain a fuzzy inverse                              for  instance  to  transform  a  matrix  into  triangular  one 
                                                                                                           through using the properties of determinant and elementary 
                 matrix and its generalized inverse. The ’fuzzy matrix’ term                               row operations. Also, a criteria is presented on the base of  
                 is the principal idea of the present paper and has more than                              Gröbner  basis  for  invertibility  of  the  fuzzy  matrix.  The 
                 two  various  meanings  in  the  research.  In  the  first  class                         organization of the paper is as follows. In Section 2, some 
                 A=(a_ij  )_(m×n)  is  said  to  be  a  fuzzy  matrix,  if                                 necessary  results  and  definitions  of  fuzzy  numbers  are 
                 a_ij∈[0,1],(i=1,2,...,m;j=1,2,...,n).  They  have  been  first                            mentioned.  Then,  the  necessary  results  and  concepts  of  
                 defined in detail and appeared with the fuzzy relations in                                Gröbner basis and polynomials are given in Section 3. In 
                 [20]. Then, there was more attention in this case [7, 18, 21].                            addition,  a  new  method  for  calculating  the  fuzzy  inverse 
                 For example, the Gödel-implication operator was used by                                   matrix  is  presented  in  Section  3.  Also,  a  criteria  and  an 
                 Hashimoto  [18]  and  he  presented  some  features  of  sub-                             algorithm  are  presented  to  find  the  fuzzy  inverse  matrix 
                 inverse  of  the  fuzzy  matrices  of  the  first  kind.  Also,  the                      when it has inverse. Section 4 containing some examples 
                 properties  of  their  regularity  were  introduced  by  Cho  in                          which  illustrate  the  algorithm.  Our  conclusions  are 
                 1999  [7].  Moreover,  a  matrix  including  entries  of  fuzzy                           summarized in Section 5.  
                 numbers is known as the fuzzy matrix, too [4, 11, 12, 19].  
                                                                                                                              II.       FUZZY BACKGROUND 
                                                                                                                 In   this    section  some  preliminaries  and  required 
                 Revised Manuscript Received on December 30, 2019.                                         background  on  fuzzy  arithmetic,  fuzzy  numbers,  fuzzy 
                 * Correspondence Author                                                                   matrices and notation of fuzzy set theory are given. 
                    Hamed  Farahani,  Department  of  Mathematics,  Chabahar  Maritime 
                 University, Chabahar, Iran.  
                                     *
                    M.  J.  Ebadi ,  Department  of  Mathematics,  Chabahar  Maritime                      Definition 2.1 [10, 23] A fuzzy number                       on the set of real 
                 University, Chabahar, Iran.                                                               numbers   is a fuzzy set if its membership function                                , 
                    Hossein  Jafari,  Department  of  Mathematics,  Chabahar  Maritime 
                 University, Chabahar, Iran.                                                                                    is as follows:  
                     
                 ©  The  Authors.  Published  by  Blue  Eyes  Intelligence  Engineering  and 
                 Sciences Publication (BEIESP). This is an open access article under the 
                 CC-BY-NC-ND license http://creativecommons.org/licenses/by-nc-nd/4.0/ 
                  
                      Retrieval Number: B6295129219/2019©BEIESP                                                  Published By
                      DOI: 10.35940/ijitee.B6295.129219                                              3030       Blue Eyes Intelligence Engineering 
                      Journal Website: www.ijitee.org                                                            & Sciences Publication 
                                                                                                                                                      
                                           Finding Inverse of a Fuzzy Matrix using Eigenvalue Method 
                                                                                     Fuzzy matrix was defined and introduced as a rectangular 
                                                                                    array  of  fuzzy  numbers  [10,  11].  Thus,  the  definition  in 
                                                                                    formal form was defined as below [9]: 
                                                                                    Definition  2.4  If  the  elements  of  a  matrix               are 
                                                                                    fuzzy numbers we say that   is a fuzzy matrix. In addition, 
             Definition 2.2 The fuzzy number           with the following           consider              and              as two fuzzy matrices of 
             function of membership            :                                    orders            and        ,  respectively.  The           is  the 
                                                                                    product  order  of  two  fuzzy  matrices  and  their  product  is 
                                                                                    given as below:  
                                                                                                                                                
                                                                                    where                               ,  where  the  approximated 
              is  called  a  number  of       type  where  the  non-increasing      multiplication denoted by       .  
             continuous  functions    and         defined  on          decrease     In  an  analogous  manner,  a  fuzzy  matrix  A  including  the 
             strictly to zero in those subintervals of           in which they      spread parts of the left and right and the center just as a 
             are satisfying the conditions                        and positive.     fuzzy number is given as the following form:                      , 
             Also, the non-negative real numbers   and   are the spread             in  which    ,     and     represent  the  center,  left  and  right 
             parameters.  Usually,        and      are  known  as  the  shape       spread matrices respectively and all are crisp. Moreover, the 
             functions. In addition, the following is parametric form of            sizes of them are the same [9]. The interested reader can 
             fuzzy numbers of type         [11]:                                    refer to [24] for further basic and essential properties of the 
                              , where left and right spreads are respectively       fuzzy matrices. 
                and  . The symmetric fuzzy number           is a fuzzy number           III.      POLYNOMIALS AND  GRÖBNER BASIS 
             in which the spreads are           [11].                               This  section  contains  the  introduction  of  some  basic 
              The operations of arithmetic were defined by Dubois and               concepts in relation to the  Gröbner basis and polynomials.  
             Prade [11] relying on the parametric representations of the            Consider       as  a  field  and             as     (algebraically 
             fuzzy numbers of         type. Here, multiplication and addition       independent)  variables.  Each  power  product                    is 
             are given for the purposes of illustration. All formulas and           called  a  monomial  where                         .  Because  of 
             more  detailed  descriptions  can  be  found  in  [11].  The           simplicity, we abbreviate such monomials by            where   is 
             addition and multiplication for two positive fuzzy numbers             used for the sequence                 and                    .  The 
             of        type                    and                   are  given     set of all monomials can be sort over        via special kinds of 
             respectively as follows:                                               total orderings which known as the orderings of monomial 
                                                                                    recalled as the below definition.  
                                                                                    Definition 3.1 The total ordering   on the monomials set is 
             and                                                                    named orderings of monomial if for each                 and     as 
                                                                                    monomials we have:   
                                                                                    •                              ,  
             It is noticeable that the outcome fuzzy number is a kind of            and  
             approximate results. In the below, we can find the scalar 
             multiplication:                                                        •    is well-ordering.  
                                                                                          There are infinitely many monomial orderings, each one 
                                                                                      
                                                                                    is convenient for a special type of problems. Between them, 
             The definitions of being positive,  negative, and zero of a            the  graded  and  pure  reverse  lexicographic  orderings 
             fuzzy number are given below:                                          represented  by              and         pointed  out  as  follows: 
                                                                                    Assume that                  . We say that   
             Definition 2.3 When               is a fuzzy number support, if 
                            the fuzzy number considered as positive. Thus,            •              whenever 
             if                 the  fuzzy  number  considered  as  negative.                                                  for an integer 
             Lastly, if                the fuzzy number considered as zero.                    .  
                  Retrieval Number: B6295129219/2019©BEIESP                              Published By
                  DOI: 10.35940/ijitee.B6295.129219                            3031     Blue Eyes Intelligence Engineering 
                  Journal Website: www.ijitee.org                                        & Sciences Publication 
                                                                          International Journal of Innovative Technology and Exploring Engineering (IJITEE)
                                                                                                                     ISSN: 2278-3075 (Online), Volume-9 Issue-2, December 2019
                     •                            if                                                                       the above polynomial system or to the ideal   is defined to 
                                                                                                                           be  
                   breaking ties when there exists an integer                                                 such 
                   that                                                                                                                                                                                                   
                                                                                                                           where   is used to denote the algebraic closure of                                     . Now, 
                          It is worth noting that the former has many theoretical                                          consider             as  a  Gröbner  basis  for    w.r.t  an  arbitrary 
                   importance while the latter speeds up the computations and                                              monomial ordering. As an interesting fact,                                              which 
                   carries fewer information out. Any polynomial on                                                        indicates that                               .  This  is  the  key  computational 
                   over             can  be  written  as                       linear  combination  of                     trick to solve a polynomial system. Let us continue by an 
                   monomials.  The  polynomial  ring  on                                                over               example.  
                   represented by                                 or just by               and is the set of               Example 3.4  We are going to solve the following 
                   all polynomials having the structure of a ring with common                                              polynomial system:  
                   polynomial  multiplication  and  addition.  The  leading 
                   monomial of   is the greatest nominal with respect to                                              
                   included  in               and  represented  by                           .  The  leading                                                                                     
                   coefficient of   is the coefficient of                                   and denoted by 
                              .Moreover,                     is said to be                                 if      is 
                   a polynomials set, and                         is said to be the initial ideal of                       By the nice properties of pure lexicographical ordering, the 
                   and the ideal generated by                               if   is an ideal. Now, we                      reduced               Gröbner                basis             of          the            ideal 
                   decide to mention the idea of Gröbner basis of polynomial                                                                                                                                            has 
                   ideal which carries lots of useful information out about the                                            the form  
                   ideal.  
                   Definition 3.2 Consider   as a monomial ordering and   as                                                                                                                                    
                   a polynomial ideal of                       . The finite set                   is said to be            with respect to                                    , where   
                   a Gröbner basis of   if for any nonzero polynomial                                               ,                              15           14            12            10         9         8
                   and for some                    ,              is divisible by                   .                       g1(z )  z                3z 5z 3z z z
                                                                                                                                         4z66z44z21
                        Using the famous basis theorem of Hilbert (See [2]), it is                                          
                   shown that any polynomial ideal holds a Gröbner basis w.r.t                                              g (z )2z14 9z1311z12 2z117z10
                                                                                                                                  2
                   any  monomial  ordering.  There  exist  also  some  efficient                                            
                                                                                                                                         3z92z8z74z67z510z4
                   algorithms to calculate Gröbner basis. The first and the most                                                                                                                                       
                   simplest one is the Buchberger algorithm which is devoted                                                                       32
                                                                                                                                          6z 11z 2z 4
                   in the same time of introducing the Gröbner basis concept                                                
                                                                                                                                                   13            12         11            10         9
                   while the most efficient known algorithm is the Faugère’s                                                g3(z )z                  3z z 2z z
                   F  algorithm [15] and another signature-based algorithms                                                              z82z62z4z33z21
                                                                                                                            
                   such as G V [16] and GVW [17]. It is worth noting that                                                   
                   Gröbner basis of an ideal is not necessarily unique. To have 
                   uniquity, we define the reduced Gröbner basis concept. We                                               This special form of Gröbner basis for this system allows us 
                   have the uniqueness of the reduced Gröbner basis of an ideal                                            to  find                by  solving  only  one  univariate  polynomial 
                   up to the monomial ordering as a significant reality.                                                             and putting the roots into the two last polynomials in 
                   Definition 3.3 Consider   as a Gröbner basis for the ideal                                                 .  
                   w.r.t.       . Then   is so called a reduced Gröbner basis of                                           Theorem 3.5  Suppose that   is a reduced  Gröbner basis 
                   whenever  each                              is    monic,  which  means  that                            for   w.r.t any monomial ordering and   is an ideal in                                         . 
                                      and for each                              none of the appearing                      If                then                   .  
                   monomials in   is divisible by                               .  
                        The help of Gröbner basis to solve a polynomial system                                                  The existence of univariate polynomials in a polynomial 
                   is one of its most applications. Consider                                                               ideal depends on the dimension of the ideal. The concept of 
                                                                                                                           dimension of an ideal is recalled in the next definition.  
                                                                                                                           Definition  3.6  Consider                        as  a  set  of  variables  and 
                                                                                                                                            as  an  ideal.  The  set  of  variables                         is  said  an 
                                                                                                                           independent set w.r.t  , whenever                                              .  
                   as  a  polynomial  system  and                                               as  the  ideal              
                   generated by                         .  The affine variety corresponding to 
                         Retrieval Number: B6295129219/2019©BEIESP                                                                Published By
                         DOI: 10.35940/ijitee.B6295.129219                                                          3032         Blue Eyes Intelligence Engineering 
                         Journal Website: www.ijitee.org                                                                          & Sciences Publication 
                                                                                                                                                                                                                           
                                                               Finding Inverse of a Fuzzy Matrix using Eigenvalue Method 
                   The  dimension  of                      is  the  maximal  independent  set                                         a basis for                  
                   cardinality w.r.t  . Furthermore, when the dimension of   is                                            for                         do 
                   zero          called  a  zero  dimensional  ideal,  and  positive                                                    the eigenvalue set of                            
                   dimensional otherwise.  Zero dimensional ideals have very 
                   nice  properties  which  facilitate  the  computations.  For                                            end for 
                   instance, for an ideal   with zero dimension, the dimension                                                                                
                   of the vector space                         is finite and one can find its basis                        for               do 
                   easily via reading the leading monomials of a Gröbnr basis.                                             if                    for an                          then 
                   A basis for                        in  which  the  set  of  all  monomials  in                                                  
                            denoted by              and can be constructed by the set                                      end if 
                                                                                                                           end for 
                                                                                                                               Return                ; 
                   More precisely, the computation of a Gröbner basis   at first                                                  Using linear algebra, eigenvalue method is a simple and 
                   is  enough  to  compute                   ,  and  carry  those  monomials  out                          efficient  method  to  solve  a  zero  dimensional  ideal. 
                   which are not divisible by                                  for each                .  A new            However,  the  result  of  cartesian  product  of  eigenvalues 
                   property of the ideals with zero dimension in described in                                              gives a superset of the solution set and so it is needed, as 
                   the  below  theorem  in  which  it  is  one  of  the  essential                                         mentioned in the algorithm, to check whether a tuple is a 
                   theorems  in  the  present  paper.  However,  the  following                                            solution or not. For instance to solve Example 3.4 by this 
                   definition should be given.                                                                             method we need to check                               tuples  to  find  out  whether 
                                                                                                                           they  belong  to  the  solution  set  or  not.  This  is  while  this 
                   Definition 3.7 Suppose that   is a polynomial ideal with                                                system  has  only                       solutions.  Thus,  this  method  is 
                   zero dimension and   is a basis for                                   . The definition                  convenient when the degree of univariate is low w.r.t the 
                   of the linear transformation                        for every polynomial                                number  of  variables.Now,  the  eigenvalue  approach 
                                   is as below:                                                                            illustrated to find the real solutions of a polynomial system 
                                                                                                                           through the below example. 
                                                                                                                           Example 4.2 [13] The following system of equations is 
                                                                                                                           considered: 
                    Also, suppose that                      is  the  matrix  representation of                        
                   w.r.t       . Therefore,              is said the multiplication matrix of   
                   w.r.t  .                                                                                                                                                                   
                                     IV. PROPOSED METHODOLOGY                                                              Suppose that   is the generated ideal via these equations. 
                                                                                                                           At  first,  a  Gröbner  basis  for    w.r.t  the  graded  reverse 
                   4.1   Eigenvalues for solving 0-D polynomial system                                                     lexicographic order is computed and so the command  
                       The following theorem and algorithm state the method of                                                                                                                                  
                   using eigenvalues to solve a zero dimensional polynomial                                                is used to find the below monomials: 
                   system.  
                   Theorem  4.1  ([2])    Using  the  above  notations,  the                                                                                                                                                  
                   eigenvalues of                 show the values of   over                        .                       Now, the matrix representation of                             w.r.t        is obtained as 
                   As a very fast conclusion of this theorem, one can solve a                                              follows: 
                   zero        dimensional             polynomial  equations  system  by 
                   calculating the eigenvalues of                             for each variable               . Of 
                   course the eigenvalues of                            are  the          th component of 
                           . 
                                   Algorithm 1. Eigenvalue Method 
                   Require:                                                               a       set        of 
                   polynomials where                                             
                   Ensure:                    
                                a  Gröbner  basis  for                          w.r.t  an  arbitrary                                                                                                                       
                   monomial ordering;                                                                                          
                         Retrieval Number: B6295129219/2019©BEIESP                                                                Published By
                         DOI: 10.35940/ijitee.B6295.129219                                                          3033         Blue Eyes Intelligence Engineering 
                         Journal Website: www.ijitee.org                                                                          & Sciences Publication 
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...International journal of innovative technology and exploring engineering ijitee issn online volume issue december finding inverse a fuzzy matrix using eigen value method hamed farahani m j ebadi hossein jafari abstract the present paper extends concept study second class is ignored because that its elements are numbers which may be complex arithmetic structure in focus implemented to model imprecise uncertain features will on this matrices investigations problems real world problem calculation invertibility square interval obtaining converted solving polynomial their two popular have been warm equations fpes system approach issues recent studies proposes an transformed equivalent crisp compute base eigenvalue solutions computed also grobner basis changing into properties criteria for introduced furthermore novel algorithm proposed find then achieving all entries regarding term order ideal at time big advantage comparing existence generated by polynomials methods end some illustrative e...

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