A GMRES Method for Solving Fuzzy Linear Equations

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1 7 Iteratoal Joural of Fuzzy Systems Vol 6 No Jue 4 A GMRES Method for Solvg Fuzzy Lear Equatos Jeyog Zhou ad Hu We Abstract Ths paper s teded to propose a method to replace the orgal fuzzy lear system by two crsp lear systems Ad the the method s mplemeted by GMRES If a fuzzy osgular lear system has a fuzzy soluto our method s able to obta the soluto Otherwse our method ca oly fd a weak fuzzy soluto At last some large scale umercal tests are preseted Keywords: Fuzzy Lear equatos strog fuzzy soluto Krylov subspace GMRES fuzzy umber vectors Itroducto I ths paper we wll cosder to solve a type of fuzzy lear equatos A b () where coeffcet A ( a ) ( ) s a crsp matr ad y are fuzzy vectors of legth Solvg fuzzy lear systems has may applcatos [9 6] For eample all or parts of lear systems arsg from mathematcs physcs statstcs ad ecoomcs etc are fuzzy umbers rather tha crsp umbers Thus t s mportat to develop umercal methods for solvg fuzzy lear equatos For ths type of fuzzy lear equatos some researchers [ ] study the estece of solutos embeddg method ad perturbato aalyss etc other researchers [ ] have studed drect methods classcal teratve methods cludg the Rchardso teratve method the Jacob teratve method the Gauss Sedel method ad the SOR method o-statoary teratve methods cludg the steepest descet method ad the cougate gradet method (CG) for osgular case of Correspodg Author: Jeyog Zhou s wth the Departmet of Appled Mathematcs Shagha Uversty of Face ad Ecoomcs Shagha 433 PR Cha ad wth the School of Appled Mathematcs Xag Uversty of Face ad Ecoomcs Urumq Xag 83 PR Cha E-mal: eyogzhou@shufeeduc Hu We s wth the School of Computer Scece Fuda Uversty Shagha 433 PR Cha E-mal: wehu@fudaeduc Mauscrpt receved May 3; accepted 6 May 4 ths type fuzzy equatos Recetly a ew approach s preseted [5] There are also may other types of fuzzy lear equatos that have bee studed [ ] It s well kow that drect methods have some shortcomg the applcato of solvg large scale ad sparse lear systems ad classcal teratve methods are ot deal choces whe the spectral radus of the coeffcet matr of () s close to Thus Krylov subspace methods are good choces for large scale ad sparse lear equatos cludg fuzzy lear systems But the studyg of Krylov subspace for fuzzy lear systems s complete because the CG method [] oly adapts to solvg fuzzy symmetrc postve defte lear systems Thus t s ecessary to develop Krylov subspace methods for usymmetrcal ad defte fuzzy lear systems Especally large scale umercal tests are eeded Ths paper s orgazed as follows I Secto we preset some otato ad defto that wll be used ths paper I Secto 3 we descrbe the Geeralzed Mmum Resdual (GMRES) method I Secto 4 ma results of ths paper are preseted I Secto 5 we preset some umercal tests I Secto 6 we dscuss strog fuzzy solutos ad weak fuzzy solutos Prelmares Fuzzy umbers troduced by Zadeh [3] are used to descrbe the vagueess ad lack of the precso of data Arthmetc operatos of fuzzy umbers were vestgated by Zadeh [ 3] Mzumoto ad Taaka [9 ] Dubos ad Prade [4] ad Nahmas [] Let us cosder a arbtrary fuzzy umber by a ordered par of fuctos ( u ( u( ) r whch satsfy the followg requremets [3]: u ( s a bouded left cotuous o-decreasg fucto over [ ] u ( s a bouded left cotuous o-decreasg fucto over [ ] 3 u( u( r A crsp umber s smply represeted by u ( u( r The set of all fuzzy umbers ( u ( u( ) becomes a cove coe that s deoted by E whch s the embedded somorphcally ad 4 TFSA

2 J Zhou ad H We: A GMRES Method for Solvg Fuzzy Lear Equatos 7 sometrcally to a Baach space [3] A trapezodal fuzzy umber u ( y ) also s a popular epresso of fuzzy umbers I ths epresso s the left fuzzess of ad s the rght fuzzess of y A fuzzy umber also has a parametrc epresso The form s gve as follows: u( r u( y r Whe y a trapezodal fuzzy umber v ( y ) v s called a tragular fuzzy umber ad epressed by v ( ) Followg [6] we preset the addto ad the scalar multplcato of fuzzy umbers by usg Zadeh's eteso prcple [3] as follows y f ad oly f ( y( ad ( y( y ( ( y( ( y( ) 3 ( k k ) k k ( k k ) k Based o those prelmares Defto of a fuzzy lear system ad ts soluto s preseted as follows Defto [6]: Cosder a lear system A y where a a a a a a A a a a s a crsp matr ad y y y y are fuzzy vectors Ths system s called a fuzzy lear system Defto [6]: A fuzzy umber vector T gve by ( ( ( ) r s called a strog fuzzy soluto or geue fuzzy soluto of () f a ( a ( y ( () a ( a ( y ( () By () ad () order to solve () oe must solve followg crsp fucto lear system CX Y (3) where C C C C C ad y X Y y I (3) X ( T ) ad T ( y y y y y y Y ) Elemets c of the matr C are determed as ad ay have a c c a c c a a c whch are ot decded are zero Thus we A C C If C s osgular crsp fucto lear system ca be uquely solved for X The fact that the matr C s osgular s coecto wth matrces B ad A Fredma eplaed the fact the followg theorem Theorem [6]: The matr C s osgular f ad oly f matrces A C C ad B C C are both osgular ad D E C E D where D ( B A ) E ( B A ) Eve f the matr C s osgular the soluto vector may stll ot be a approprate fuzzy vector The followg theorem provdes ecessary ad suffcet codtos for the uque soluto vector to be a fuzzy vector Theorem [6]: Let C be osgular the uque soluto of (3) s always a fuzzy umber vector for arbtrary vector Y f ad oly f C s oegatve Remark : By Theorem the requremet that strog fuzzy soluto of () for arbtrary Y s equvalet to the requremet that verse matr of C s oegatve But for some gve Y the strog fuzzy soluto of () may est eve f oegatve Eample 3 [6] demostrates ths fact Thus codtos Theorem are ot eeded for the estece of strog fuzzy soluto of osgular fuzzy lear equatos If X s a soluto of system (3) but s ot a fuzzy umber vector s a weak fuzzy soluto of system () for ths stuato At last we preset the defto of fuzzy solutos whch covers deftos of strog fuzzy soluto ad weak fuzzy soluto gve by Fredma [6] C s ot

3 7 Iteratoal Joural of Fuzzy Systems Vol 6 No Jue 4 Defto3 [6]: Let X {( ( ( ) } deote the uque soluto of the system (3) The fussy umber vector u {( u ( u ( ) } defed by u m{( ( ( ())} ( u ( ma{( ( ( ())} s called the fuzzy soluto of system () 3 Geeralzed Mmum Resdual Method The Geeralzed Mmum Resdual (GMRES) [3] method s a well-kow Krylov subspace teratve method for solvg large scale sparse osgular crsp lear systems A b It has the property that ths resdual orm ca be computed wthout the terato havg bee formed Thus the epesve acto of formg the terato ca be postpoed utl the resdual orm s deemed small eough Just lke Cougate Gradet (CG) method t forms a orthogoal bass for the Krylov subspace m km spa{ r Ar A r } where ad s a tal guess for the method However t forms ths bass eplctly by modfed Gram-Schmdt orthogoalzato method We preset ths method as follows Algorthm 3 GMRES Compute r b A r ad v : r / Defe the ( m ) m matr H ( h ) m m Set H For m Do: Compute w : Av For Do: h : ( w v ) w : w hv EdDo h w If h set m : ad go to v w / h EdDo (m) Compute y the mmzer of e H m y ad ( m) V ( m) y where m e Whe m s large the GMRES method s mpractcal because of the growth of the memory ad the lmted storage So we take the restart trck to overcome ths problem Gve a restart parameter m whe GMRES terates m steps we set the m-th appromate soluto (m) as the tal teratve soluto for the et recycle ) ( ) e ( m The detal of practcal mplemetato ca be foud [3] 4 Decreasg Matr s Scale I ths secto we preset a useful Theorem () () Theorem 4: Let C ( c ) ad C ( c ) be real matrces For the matr C C C C C there ests a orthogoal matr I I Q I I such that Q C C A T CQ C C A where I s a I detty matr Proof The proof s drect Let I J ad Q ad recall that C ( A A ) ad C ( A A) the the matrces C ad Q could be wrtte as C I C J C ad Q Q I After drect calculato we get followg corollary Corollary : Theorem s a corollary of Theorem 4 By Theorem 4 t s uecessary to solve the y eteded fucto lear system CX Y to get y solutos of the fuzzy lear systems A y We ust solve two lear crsp fucto systems ( C C) X Y ad ( C C) X Y ad the soluto s ( ) X ( y y) Y X ( ) Y ( y y)

4 J Zhou ad H We: A GMRES Method for Solvg Fuzzy Lear Equatos 73 Actually A C C ad A C C Thus we ca compute A X Y (4) AX Y (4) respectvely or parallel for all r [] Algorthm 4: Compute (4) ad (4) parallel Compute X ad X Remark 4: The requremet that C ests s equvalet to the requremet that both A ad A are vertble However the ecessty of kowg a pror that A ests s delcate Thus whe A s vertble ad A s ot vertble our method caot be used to solve () Because GMRES wll compute a least squares soluto eve f A s sgular But strog fuzzy soluto or weak fuzzy soluto may est Please cosder followg eample Eample 4: ( r (4 r7 Let A Clearly A s sgular So C also s sgular By usg Algorthm (4) we ca get the followg soluto (335 5 (5 75r35 5 Actually sce A the true soluto s ( r ( r Remark 4: I [5] aother approach to replace the orgal fuzzy lear system by two crsp lear systems s preseted It eed to solve A( ) y y frst ad to solve A y Cd or A y Cd et where c Thus the method ca ot mplemeted parallel ad eed more multplcato operatos 5 Numercal Tests I ths secto we preset some umercal tests for our algorthm All of our tests are mplemeted a IBM computer wth 5 Memory (DDR) ad double 6G (Itel) CPUs We wll compute a fuzzy lear system A b by selectg A as a Posso matr [8] The Posso matr s a block matr D I I D I A I D I I D where tur the matr D s a trdagoal matr wth costat dagoal 4 ad sub-dagoals of ad I s a detty matr Thus the matr A s a ad C s a matr Ad the we costruct a osgular fuzzy lear system wth b As where s ( r e e s a crsp vector of legth We use the Jacob teratve method the Gauss-Sedel method the SOR method (see [4 9]) GMRES for (3) ad the Algorthm 4 to compute ths eample I whose etres are all oes our tests we select a zero vector as tal teratve vector X Ad the covergece crtero s tol X ( k ) S S 6 We frst test the eample wth 6 For the Jacob teratve method t takes 749 secods ad teratve steps to coverge For the SOR teratve method we choose the rela parameter w to solve the eample The computg results are lsted Table Table SOR for =6 w teratve steps CPU tme (secods) (s) (s) (s) (s) 5 34 (s) (s) (s) 9 4 3(s) We fd that whe the value of w s close to 7 we have a less epesve computg however whe the value of w s far from 7 computg costs crease Thus the optmum value of w s almost 7 Ad we also fd that for most value of w computg costs are epesve For the Gauss-Sedel teratve method we cost 46

5 74 Iteratoal Joural of Fuzzy Systems Vol 6 No Jue 4 teratve steps ad secods to coverge 7 For Algorthm 4 accuracy of for (4) ad (4) are eeded for covergece Computg results of Algorthm 4 ad GMRES for (3) are lsted Table The values the secod colum of ths table are outer (e teratve steps of GMRES For eample 9(6) the frst row of Table meas the GMRES to take 9 outer teratos ad addtoal 6 er terato to coverge Obvously the whole computg costs of the Algorthm 4 are much cheaper Table Computg results for 4 restart 4 4 CPU tme GMRES CPU tme parameter (secods) step for (secods) of for 4 CX = Y GMRES for CX = Y 3(6) 9(6) 5(s) 4 79 (4) 3(7) 78(s) (7) (3) (9) 65(s) (5) 94 More drectly the computg results of the Jacob teratve method the Gauss-Sedel teratve method the SOR method ( w 7 ) ad the Algorthm 4 mplemeted parallel by GMRES are depcted the Fgure Whe = 3 ths eample we get a fuzzy lear system wth 4 varables The Jacob teratve method does ot coverge after teratg steps However the Gauss-Sedel teratve method costs 53 teratve steps ad 8664 secods to coverge For the SOR method whe w 6 the algorthm coverges very slowly Thus we test the SOR method for w The computg results are lsted Table 3 Clearly the optmum value of w s 8 ad the computg costs of the SOR are ot cheap Table 3 SOR for =3 w teratve steps CPU tme (secods) (s) (s) (s) (s) 5 34 (s) (s) (s) 9 4 3(s) For Algorthm 4 ad GMRES for (3) we lst computg results of 4 ad 4 Table 4 We could draw a cocluso that parallel or respectvely mplemetg Algorthm 4 s much faster tha other teratve methods our paper Moreover f we take optmal restart parameters for (4) ad (4) less tha secods for parallel mplemetato s eeded More drectly we descrbed the computg results of the Algorthm 4 mplemeted by GMRES (5) for (4) ad (4) parallel the SOR (w = 8) the Jacob teratve method ad the Gauss-Sedel teratve method Fg restart parameter lg( X S / S ) Table 4 Computg results for 4 GMRES SOR JACOBI GS teratve step Fgure The vector S s the fuzzy soluto GMRES meas Algorthm 4 mplemeted by GMRES for (4) ad (4) parallel lg( X S )/ S CPU tme (secods) for 4 GMRES step for C X = Y 4(9) 33(7) 463(s) 9(6) 65 (7) 8(7) 73(s) 4(3) (8) 5(8) 5938(s) 3(8) (37) 3(3) 565(s) (4) (37) (3) 38(s) (48) 3398 CPU tme (secods) of GMRES for C X = Y 6 (37) () 453(s) (47) 3536 GMRES SOR JACOBI GS teratve step Fgure The vector S s the soluto ad the curve of GMRES meas mplemetg Algorthm 4 parallel by GMRES (5)

6 J Zhou ad H We: A GMRES Method for Solvg Fuzzy Lear Equatos 75 6 Strog Fuzzy Soluto or Weak Fuzzy Soluto Whe fuzzy lear systems are osgular a geue fuzzy vector soluto must cota all fuzzy umbers that satsfy three codtos defed at the begg of Secto Accordg to Theorem matr C should be oegatve to esure the uque soluto s a geue fuzzy soluto for a arbtrary rght had Ad the oegatve verse of a oegatve matr meas that ths matr s a geeralzed permutato matr It s well kow that to use GMRES o permutato matrces ca be effcet ad covergece could be acheved oly after -steps However by Remark for a gve rght had a geue fuzzy soluto may est eve f C s ot oegatve Therefore may fuzzy lear systems wth C ot beg a geeralzed permutato matrces have geue or strog fuzzy solutos [6] other ust have weak fuzzy solutos [6] Obvously we ca obta a strog fuzzy soluto by employg our algorthms f a fuzzy lear system has a strog fuzzy soluto Otherwse we oly ca obta a weak fuzzy soluto by employg our algorthms Moreover we wll take the followg eamples to accout for weak fuzzy solutos or strog fuzzy solutos Both Eample 6 ad Eample 6 come from [6] Eample 6: For the followg fuzzy system ( r 3 (4 r7 ts eteded crsp lear system s ( r 3 ( 4 r ( ) r r 3 ( r 7 We employ Algorthm 4 to obta followg solutos ( r ( r ( r ( 875 5r Obvously these solutos are strog fuzzy solutos Eample 6: Let 3 ( r 3 ( r3) 33 ( Its eteded crsp lear system s ( r ( r 3 3( ( r ( 3 3 3( r We employ Algorthm 4 to obta the followg soluto ( r ( r ( r 3 ( r ( r 3( r Obvously ad are ot fuzzy umbers [6] So we ca get weak fuzzy solutos u ( r u ( r u3 ( r Cocludg Remarks I ths paper a GMRES method for osgular fuzzy lear systems s preseted Whe a osgular fuzzy lear system has a strog fuzzy soluto Algorthm 4 ca be employed to get ths soluto fast Otherwse Algorthm 4 oly ca fd a weak fuzzy soluto Some large scale umercal tests also show that Algorthm 4 s much faster tha other teratve methods Refereces [] S Abbasbady A Jafara ad R Ezzat Cougate gradet method for fuzzy symmetrc postve defte system of lear equatos Appl Math Comput vol 7 pp [] S Abbasbady R Ezzat ad A Jafara LU decomposto method for solvg fuzzy system of lear equatos Appl Math Comput vol 7 pp [3] S Abbasbady ad A Jafara Steepest descet method for system of fuzzy lear equatos Appl Math Comput vol 75 pp [4] T Allahvraloo Successve over relaato teratve method for fuzzy system of lear equatos Appl Math Comput vol 6 pp [5] T Allahvraloo A commet o fuzzy lear

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