Iterative Methods for Solving Fuzzy Linear Systems

Transcription

1 Astrala Joral of Basc ad Appled Sceces, 5(7): , 2011 ISSN Iteratve Methods for Solvg Fzzy Lear Systems 1,2 P. Masor, 1 B. Asady 1 Departmet of Mathematcs, Arak Brach, Islamc Azad Uversty, Arak, Ira. 2 Departmet of Compter Scece, Delh Uversty, New Delh, Ida. Abstract: Lear systems of eqatos, wth certaty o the parameters, play a maor role several applcatos varos areas sch as ecoomcs, face, egeerg ad physcs. I ths paper, we proposed a method for solvg a fzzy lear system to form AX =b. I whch A s osglar crsp matrx ad b s arbtrary fzzy vector. Frst we solved a fzzy traglar system ad by sg of LU-Decomposto ad/or Gassa Elmato methods exted ther methods for solvg a geeral fzzy lear system. Some examples are preseted to llstrate these methods ad compared wth others methods. Key words: Fzzy mber, Fzzy lear systems, Gassa Elmato Method, LU-Decomposto, caocal form. INTRODUCTION Lear system of eqatos play a maor role may of areas of scece sch as egeerg, mathematcs, physcs ad ecoomcs. I may practcal problems at least some of the system s parameters ad measremets are fzzy rather tha crsp. Aalyzg sch systems reqres the se of fzzy formato. Therefore, t s mmesely mportat to develop mathematcal models ad mercal procedres that wold approprately treat fzzy lear system ad solve them. Lear system AX= b, where the elemets a of the matrx A ad the elemets b of the vector b are fzzy, s called a fzzy lear system. Bckley (1991) proposed a method for solvg t ad S. Mzzol ad H. Reyaerts, (2006) geeralzed ths method for solvg the fzzy lear system of the form A 1 X b 1 = A 2 X b 2. Sch R. Horcík, ad A. Vroma ad co worker, (2007) solved t wth Cramer's rle. Bt, tme of compte for solve a problem by Cramer's rle s excessve. A partclar type of fzzy system AX =b whch the coeffcet matrx A s crsp ad b a fzzy vector vestgated by may wrters sch as (Abbasbady et al., 2006; Allahvraloo, 2005a; 2005b; 2003; Asady et al., 2005; 2009; Fredma, et al., 1998; 2000; Mzmoto, 1979; Nahmas, 1978; Nasser ad Ardl, 2005; Zheg ad Wag, 2006) So that for solvg t (Abbasbady et al., 2006; Allahvraloo, 2005a; 2005b; 2003; Asady et al., 2005; Fredma, et al., 1998; 2000; Zheg ad Wag, 2006), they se embeddg method gve (Cog-X ad Mg, 1992) ad replaced the orgal fzzy lear system by a (2) (2) crsp fcto lear system. Bt fortately, sometmes ther methods are able to solve fzzy lear system, sce the (2) (2) coeffcet matrx may be sglar eve f the coeffcet matrx A be osglar (see followg example ). Example 1.: Cosder the 2 2 fzzy lear system as follows x1 x2 = y1, x1 x2 = y2. So that the matrx A s osglar, whle the matrx 2 2 Correspodg Athor: B. Asady, Departmet of Mathematcs, Arak Brach, Islamc Azad Uversty, Arak, Ira. E-mal: b-asad@a-arak.ac.r, babakmz2002@yahoo.com. 1036

2 Ast. J. Basc & Appl. Sc., 5(7): , S =, s sglar. I others word, for some of lear eqatos, the proposed methods (Abbasbady et al., 2006; Allahvraloo, 2005a; 2005b; 2003; Asady et al., 2005; Fredma, et al., 1998; 2000) ca ot reach a solto or they offer a fte mbers of soltos. I ths paper we propose LU-Decomposto method for solvg a ( ) fzzy lear system becase traglar LU-Decomposto method s mportat for great practcal to solvg systems of lear eqato. Therefore, Ths process wold be developed throgh sx sectos. I secto 2, we recall some fdametal reslts o fzzy mbers ad defe fzzy lear system. I secto 3, we propose a method for solvg a fzzy lear traglar system ad Gassa elmato method ad LU-Decomposto Method secto 4. Nmercal examples are proposed secto 5 ad last secto s devoted to coclsos. 2 Prelmares: Defto 1.: A arbtrary fzzy mber s represeted by a ordered par of fctos parametrc form s a par (, ) of fcto r (), r (),0r1, whch satsfes the followg reqremets: r () s a boded left cotos odecreasg fcto over [0,1], r () s a boded rght cotos ocreasg fcto over [0,1], r () r (),0r1. The set of all these fzzy mbers s deoted by E. Defto 2.: The lear system a11x1 a12x2 a1 x = b1, a21x1 a22x2 a2x = b2, a 1x1 a2x2 ax = b, (1) Or followg matrx form Ax = b (2) where the coeffcet matrx A =( a ),1 ad 1 s crsp matrx ad b E, 1 s called a fzzy system of lear eqatos(fls). For arbtrary fzzy mbers x = ( xr ( ), xr ( )), y= ( yr ( ), yr ( )) ad real mber k, we may defe the addto ad the scalar mltplcato of fzzy mbers by sg the exteso prcple as 1037

3 Ast. J. Basc & Appl. Sc., 5(7): , 2011 x y=(() xr yr (), xr () yr ()) kxx (, )=( kxkx, ), k 0, kxx (, )=( kxkx, ), k. (3) 3 Solto of Fzzy Lear Systems: Bckley ad Q, (1991) have bee proposed a method to solve a fzzy lear system Ax = b, where the elemets, a, of the matrx A ad the elemets, b, of vector b are fzzy mbers. They defe the solto by, for all r0 [0,1] ()={ ( r t a )= t y, a [ a (), r a ()], r y [ b (), r b ()]}, r ad for all B t R x ( t) = sp r r[0,1], t( r). Clearly, f A was be a crsp matrx the we have ()={ r t At = y, y [ b (), r b ()]}, r for 0r 1. We see that x B s defed as a fzzy set o R ad ot as a vector of fzzy mbers see. (Horcík) Therefore, x B expresses to what extet the real vector x s a solto of the system of lear eqato, Ax = b. We prefer to defe a solto as a vector of fzzy mbers. I order to let s trodce or solto by parametrc form to every compoet of the solto vector,e. () x ()=( r x (), r xb ()) r B where B x ( r) = m{ x x( r) ad x s compoet of vector x}, B x ( r) = max{ x x( r) ad x s compoet of vector x}, B (5) (6) for all {1,2,..., }. Now we have bee proposed a method for solvg of fzzy traglar lear system (FTLS) to form Ux = b (7) whch U s a crsp traglar matrx sch that compoets of dagoal are ozero ( 0, = 1,2,..., ) ad b a arbtrary fzzy vector. Lemma 1: If U (FTLS) (7) s a pper traglar matrx the, parametrc form of the compoets solto s same as followg x =m{ b /, b/ }, x =max{ b /, b/ }, x =m1/ { b ( x x ), b ( x x )}, k kk k k x =max1/ { b ( x x ), b ( x x )}. k kk k k 1038

4 Ast. J. Basc & Appl. Sc., 5(7): , 2011 I whch k = -1, -2,...,2,1. Proof.: Proof for x s obvos. Bt for x k where k = -1,...,1 sce U s a pper traglar matrx also we sppose x, x 1,..., xk 1 are kow. Therefore, for ay t() r k th compoet of ts as follow = t = y t, kk k k = k1. b () r y b () r x () r t x () r =, 1, 2,..., 1 whch k ad for ay k. Hece, k k b(), r x () r are mootocally decreasg ad b, x ( r) are mootocally creasg for all 1, k 1 therefore, we have kktk bk x x >0 ad kktk bk x x ad or eqalty t m1/ { b ( x x ), b ( x x )}, k kk k k t max1/ { b ( x x ), b ( x x )}. k kk k k Sch t be arbtrary also, accordg to eqatos (5) ad (6) we have x =m1/ { b ( x x ), b ( x x )}, k kk k k x =max1/ { b ( x x ), b ( x x )}. k kk k k Lemma 2.: If U fzzy lear system (7) (Ux = b) s a lower traglar matrx the, for x = x ( r), x 1 1 1( r) we have x =m{ b /, b / }, x =max{ b /, b / } x = x (), r x (), r k =2,3,..., ad for k k k x =m1/ { b ( x x ), b ( x x )} k kk k k x =max1/ { b ( x x ), b ( x x )} k kk k k Proof.: Proof s same as lemma 3. If U (FTLS) (7) s a pper traglar matrx the, We ca easly solve t by back sbsttto. Becase 1039

5 Ast. J. Basc & Appl. Sc., 5(7): , therefore we obta the x from th eqato as follow x =m{ b /, b/ }, x =max{ b /, b/ }. Wth the kow vale of x =( x, x ) ad sbsttted to the ( 1) th eqato, solve ( 1) th eqato based o relatos (4), (5)ad lemma 1, we obta x -1. We proceed the same way, obtag x,..., x 2 1, oe at a tme ad ths order or eqalty for k = 1, 2,...,1 we get x =m1/ { b ( x x ), b ( x x )} k kk k k ad x =max1/ { b ( x x ), b ( x x )}. k kk k k The same deas ca be exploted to solve a system that have a lower traglar strctre ad s called forward sbsttto. I ext secto we trodce two methods for solvg fzzy lear system (1). 4 Itratve Methods: I ths secto, we try exted dea of secto 3 for solvg a geeral fzzy lear system (??) by Gassa elmato ad LU-decomposto methods. 4.1 Gassa Elmato Method: I ths sbsecto we wat proposed Gassa Elmato method for solvg fzzy lear system (1). Sce coffctos matrx the system (1) s crsp ad rght had sde s a arbtrary fzzy vector the, we ca defe modfed addtoal matrx as follows: [ A b] = [ A ( b, b)], (8) or eqalty a11 a12 a1 ( b1, b1 ) a a a ( b, b ) a 1 a2 a ( b, b) [ Ab ]=, (9) We sg of trasformed steps by approprate rearragemet ad lear combatos of matrx (9) obtaed the solto from fzzy lear system 1 (see algorthm 1 Appedx). I other words, we mltpled frst row to a / a 1 11 a = a ( a / a ) a (1) ad for last colm f (1) = ( 1/ 11) 1, b b a a b ad (1) 1 11 b = b ( a / a ) b, ad addto to th row, based o eqato () as follows. a / a the

6 Ast. J. Basc & Appl. Sc., 5(7): , 2011 else f a / a > (1) 1 11, the b = b ( a / a ) b ad (1) = ( 1/ 11) 1, b b a a b 1 for = 2,3,...,. Coseqetly we have a =0 1 for 2 ad or a11 a12 a1 ( b1, b1 ) (1) (1) (1) (1) (1) (1) 0 a22 a2 ( b2, b2 ) [ A b ]=, (1) (1) (1) (1) 0 a2 a ( b, b ) fally cotg ths rearragemet, we wll have a11 a12 a1 ( b1, b1) (1) (1) (1) (1) ( 1) ( 1) 0 a22 a2 ( b2, b2 ) [ A b ]=. ( 1) ( 1) ( 1) 0 0 a ( b, b ) Therefore, for solvg lear system (1) eqalty we solve followg lear system Ux = z (10) whch 1 1 a11 a12 a (, ) 1 b b (1) (1) (1) (1) 0 a22 a 2 ( b2, b2 ) U =, z = ( 1) 0 a ( 1) ( 1) ( b, b ) Now, It ca be solved easly by " back sbsttto " based o sad rle secto 3. Clearly fzzy solto of system (10) applcable to system (1)(see theorem 1). Theorem 1: Let AX= b s fzzy lear system. If by sg exteso of Gassa Elmato method, we trasformed steps by approprate rearragemet ad lear combatos of matrx [ A b] ad reslt of ths trasformed wll be pper traglar fzzy system Ux= y. The these systems are same solto. Proof.: Proof s as prove of Gassa Elmato theorem (Datta, 1995) 1041

7 Ast. J. Basc & Appl. Sc., 5(7): , LU-Decomposto Method: I ths sbsecto we propose LU-Decomposto method for replacg coeffcet crsp matrx A by LU system (1), that L s t osglar matrx ad U s pper traglar matrx. Theorem 2: Sppose A s osglar matrx, The A has a qe decomposto: A = LU where L s t lower traglar ad U s pper traglar.(datta, 1995). (10) Theorem 3.: Let A0C be a symmetrc postve defte matrx the there exsts a qe lower traglar matrx L wth postve dagoal etres sch that A = LL t (Golb ad VaLoa, 1984). Now sppose that matrx A s osglar the, we ca wrte system (??) to below form Lx= b (11) To solve ths assme that Ux = z the Lz = b, for obta fzzy solto of fzzy lear system (12), frst we mst solvg fzzy lower traglar system Lz = b ad the pttg solto of ths system to pper traglar system Ux = z ad solvg ths system. Or z1 = b1, l21z1 z22 = b2, lk1z1lk2z2... lkk 1zk1zk = bk, l11z1 l 12z2 l 12z2 z 1 = b 1, l1z1 l2z2 l 1z1 z = b. (12) We solvg lower traglar system (12) by forward sbsttto ad obta fzzy solto as follows ( z, z )=( b, b ) ad for k = 2,..., 1 ad = 1,..., k 1 z = b ( l z l z ), k k >0 zk = bk ( l z l z ). By pttg fzzy vector fzzy lear system to below form z z1 z2 z = (,,..., ) t to pper fzzy lear system Ux = z, prodce pper 1042

8 Ast. J. Basc & Appl. Sc., 5(7): , x1 12x2 1 x = z1, 22x2 2x = z2, kk xk kx = zk, x = z, (13) by sg sad rle secto 3 we ca obta fzzy solto of pper traglar fzzy lear system (13) or eqalty, we gve ( x, x)=1/ ( z, z) ad for k = 1, 2,...,1ad =, 1,..., k1 ( xk, xk)=1/ kk zk ( x x ), r >0 zk ( x x ). Clearly, ths solto applcable to system (1). A smple algorthm to trasform a geeral matrx A to LU-Decomposto s gve appedx. 5 The examples: Example 1.: Cosder the 3 3 fzzy system 4 x1 x2 x3 = ( r,2 r), x1 3 x2 x3 = (2 r,3), 2x1 x2 3 x3 = ( 2, 1 r), let Frst, we solvg ths system by sg Gassa Elmato. (15) ( r,2 r) [ Ab ]= (2 r,3) ( 2, 1 r) we trasformed steps by approprate rearragemet ad lear combatos of matrx ths trasformed s [ A b] ad reslt of 1043

9 Ast. J. Basc & Appl. Sc., 5(7): , ( r, 2 r) ( r, r) ( r, r) By back sbsttto, we solvg pper traglar fzzy system Ux = z, where ( r,2 r) U = , z = ( r, r) ( r, r) we obta fzzy solto of system (15) ( r, r) xg ( r) = ( r, r) ( r, r) If we solvg ths system by sg LU-Decomposto method, we have A = = Frst, we solvg lower traglar fzzy lear system z1 ( r, 2 r) z = (2 r,3) z 3 ( 2, 1 r) By forward sbsttto, we obta fzzy solto: z = (( r,2 r),( r, r),( r, r)) fzzy solto of Eq.(15)wll be obta by solvg pper fzzy lear system: x1 ( r, 2 r) x 2 = ( r, r) x 3 ( r, r) The, we have ( r, r), xlu ( r) = ( r, r), ( r, r) ad soltos are same as. I example(2), we compare or method wth methods (Abbasbady et al., 2006; Allahvraloo 2005a; 2005b). 1044

10 Example 2: Cosder the 3 3 fzzy system Ast. J. Basc & Appl. Sc., 5(7): , x1 x2 x3 = (118 r,27 8 r), x1 x2 x3 = ( r, 5 8 r), 2 x1 x2 x3 = (105 r,2712 r), Frst, we solvg ths system by sg Gassa Elmato. let ( 118 r, 278 r) [ Ab ]= ( r, 5 8 r) ( 10 5 r, r) we trasformed steps by approprate rearragemet ad lear combatos of matrx of ths trasformed s [ A b] (16) ad reslt ( 118 r, 27 8 r) ( r, r) ( r, r) By back sbsttto, we solvg pper traglar fzzy system Ux = z, where ( 118 r, 27 8 r) U= , z = ( r, r) ( r, r) we obta fzzy solto of system (15) ( r, r), xg ( r) = ( r, r), ( r, r). If we solvg ths system by sg LU-D ecomposto method, we have A = = Frst, we solvg lower traglar fzzy lear system z1 ( 118 r, 27 8 r) z = ( r, 5 8 r) z 3 (10 5 r, r) By forward sbsttto, we obta fzzy solto: 1045

11 Ast. J. Basc & Appl. Sc., 5(7): , 2011 z = (( 118 r,27 8 r),( r, r),( r, r)) fzzy solto of Eq.(17)wll be obta by solvg pper fzzy lear system: x ( 118 r, 278 r), x 2 = ( r, r), x 3 ( r, r) The, we have ( r, r), xlu ( r) = ( r, r),. ( r, r) Clearly ths example Rak (A) = 3 ad A s osglar bt the 6 6 crsp matrx S followg form S = s sglar (Rak (S) = 5) ad proposed methods (Abbasbady et al., 2006; Allahvraloo 2005a; 2005b) ca ot solvg ths system. Sce S s sglar (Fredma et al., 1998). Coclso: I ths paper, we are vestgated two methods for solvg of fzzy lear system ad compared wth other methods. Also proposed theorems ad remarks for exstece of a solto qe. We are gve some examples to showg the effcecy of the proposed schemes. Sch we table 1 show, Comptg tme of or method s 1/3 as compared wth others methods. Therefore, whe wll be very large, comptg tme of or methods s very less tha related to other methods preseted (Abbasbady et al., 2006; Allahvraloo, 2005a; 2005b; 2003; Asady et al., 2005; 20009; Fredma, et al., 1998; 2000; Mzmoto; 1979; Nahmas, 1978; Nasser ad Ardl, 2005) ad t s very effcet for solve a large lear system ad exteso to very large system. Table 1: Comparatve of tme of methods Methods Mltplcato Smmato Abbasbady Allahvraloo Dehgha r method Appedx: Algorthm 1. Gassa Elmato: Ipt: [ A: b] 1 C s osglar matrx. Otpt: U s pper traglar matrx, where 0, = 1,..., ad c s a 1 fzzy vector sch that 1046

12 Ast. J. Basc & Appl. Sc., 5(7): , 2011 soltos of two systems Ax = b ad Ux = c are same. for s= 1: do = a 1s 1s f (s>1) the r s1 =0 eddo ( c, c )=( b, b ) for =1: 1do r = a,( c, c )=( b, b ) 1 for = 1: do for k = : do = a ( a / a ) a k k k eddo f ( a / a )>0 the c =m{ b ( a / a ) b, b ( a / a ) b }, c =max{ b ( a / a ) b, b ( a / a ) b } ese c =m{ b ( a / a ) b, b ( a / a ) b }, c =max{ b ( a / a ) b, b ( a / a ) b} eddo ed do prt(u, c ) eddo Recrrece algorthm for solvg ay traglar fzzy lear system,wth ozero dagoal of coeffcet traglar matrx s gve by the followg. Algorthm 2. Recrrece Algorthm for Solvg Traglar Fzzy Systems: A C Ipt: s osglar matrx,b s 1 fzzy vector s gve to system (1).. Otpt: s 1 fzzy solto that satsfy to fzzy lear system Ax = b. LU Decomposto( A, method = GassaElmato) z, z )=( b, b ) for = 2: do SUM1=0; SUM 2=0 1047

13 Ast. J. Basc & Appl. Sc., 5(7): , 2011 for =1; -1 do f ( l < 0) the SUM1 = SUM1 l z SUM 2= SUM 2 l z else SUM1 = SUM1 l z SUM 2= SUM 2 l z eddo ( z, z )=( b SUM1, b SUM2) prt z =( z, z ) eddo f ( >0) the ( x, x ) = 1/ ( z, z ) else ( x, x ) = 1/ ( z, z ) for = -1 : 1 do SUM 3=0, SUM 4=0 for = +1 : do f (( > 0) the SUM 3= SUM 3 x SUM 4= SUM 4 x else SUM 3= SUM 3 x SUM 4= SUM 4 x eddo ( x, x ) = 1/ ({ z SUM 3},{ z SUM 4}) prt x =( x, x ) eddo REFERENCES Abbasbady, S., R. Ezzat, A. Jafara, 2006.LU decomposto method for solvg fzzy system of lear eqatos,appl Math Compt., 172: Allahvraloo, T., Solto of a fzzy system of lear eqato, Appl Math Compt., 155: Allahvraloo, T., Sccessve over relaxato teratve method for fzzy system of lear eqatos, Appl Math Compt., 62: Allahvraloo, T., A commet of fzzy system of lear eqatos, Fzzy Sets ad Systems., 140: 554. Asady, B., S. Abbasbady, M. Alav, Fzzy geeral lear systems, Appl Math Compt., 169: Asady, B., P. Masor, Nmercal Solto of Fzzy Lear System, Iteratoal Joral of Compter Mathematcs., 86:

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