A New Algebraic Method for Computing the Pfaffian González, H.E. 1, and Carmona, L., J.J. 2 1 Information Technology Department, ININ, ITTLA

Transcription

1 bstrct ths pper we postlte ew decoposto theore of trx R to two trces ely lower trglr oe ll whose etres re deterts d pper trglr trx U whose etres re lso detert for pplyg ths ew ler trsforto to sew-syetrc trx R by es of prtl pvotg strtegy we obt the ff of fro the prcpl dgol of trx U ely: l l l Frtherore f we pply ths ew ler trsforto ord verso wth the bove pvot strtegy to the geted trx R we obt ff djgte G trx Wth t we c solve systes of sew-syetrc ler eqtos geerl logosly to the Crer s Rle whch c be see s rto of two ffs Both lgorths preset O copttol coplexty dex ers ff LU decoposto ff-crer rle ff-djgte trx NRODUCON he detert of sew-syetrc trx s the sqre of other expresso whch s clled the ff [] For exple New lgebrc ethod for Coptg the ff Gozález HE d Cro L forto echology Deprtet NN L forto echology Deprtet NN hs property ws proved by Cyley 89 [] he detert of sew-syetrc trx of odd order s lwys Recetly Kth [] short hstory of ffs rges tht these re soe wy ore fdetl th deterts to whch they re closely relted oh Fredrch ff trodced the fctos tht ow ber hs e 85 [ pp 96-] whle developg geerl ethod to solve systes of frst-order prtl dfferetl eqtos He gve two procedres for lstg ll perfect tchg s sbset of edges whch covers ll vertces of the grph d observed tht whe the tchg re ordered lexcogrphclly the correspodg sgs re strctly ltertg Forlly speg the ff of sew-syetrc trx R c be defed s follows []: sg weght perfect tchgs Here perfect tchg wth j j j edges s wrtte s by coveto j for ech he sg of the tchg s defed s the sg of j j j Whe ths s regrded s pertto of he weght of s j j j Becse of ther close coecto wth tchg s ffs re of cosderble terest cobtory lyss s see [56] For exple for cert grphs cldg the plr oes t s possble to cot the ber of perfect tchg s polyol te by sg ffs he relto Shows tht deterts re jst specl cses of ffs tht correspod to bprtte grphs ths sese the ff s ore bsc oto whch wold deserve ore thorogh derstdg despte the trdtol prevlece of the detert the crrcl d pplctos hs wor expects to cotrbte tht drecto Ler Syste of Eqtos LSE c be defed s set of eqtos wth ows represeted by trx vector b d ow vector x ely x b he Gss rsforto G for solvg LSE hs proved to be the best opto for ost prctcl pplctos he ew trsforto proposed here c be obted fro t Next we brefly revew ths topc LU- RCL DECOOSON WH G NOON ND DEFNONS he proble of solvg ler syste of eqtos x b s cetrl to the feld of trx coptto here re 8

2 severl wys to perfor the elto process ecessry for ts trx trglto We wll focs o the Doolttle-Gss elto ethod: the lgorth of choce whe s sqre dese d -strctred x Let s sse tht R s osglr d tht we wsh to solve the ler syste x b Here we show how for exct rthetc d prtl pvotg d col terchges soe Gss trsfortos c U erly lwys be fod sch tht s pper trglr [7] he orgl x b proble s the eqvlet to the pper trglr syste Ux b bc-sbsttto whch c be solved throgh x Sppose the tht R d tht for soe < We hve detered the Gss trsfortos x R Sch tht where: s pper trglr trx Now f d the the ltplctors : defed wth re well So we hve the followg Defto eleetry lower trglr trx of order d dex s trx of the for [8] where: e e tes geerl eleetry lower trglr trx hs the bove for he copttol sgfcce of eleetry lower trglr trces s tht they c be sed to trodce zero copoets to vector hs he trx s sd to be G he vector s referred to s the Gss vector he copoets of re ow s ltplers he t follows tht s pper trglr trx hs process llstrtes the -th step of the decoposto process whch we sed e e We fd the fl expresso for the decoposto process s L L L L geerl the forwrd elto cossts of - steps t the -th step ltples of the -th eqtos re sbtrcted fro the reg eqtos to elte the -th vrble f the pvot eleet s ll or sffcetly sll t s dvsble to terchge eqtos before ths s doe throgh pertto trx tht records the row exchges s detled below LU DECOOSON HEORE Usg the bove expresso the followg c be estblshed [9]: heore Let deote the leder or sb-trx x x of R f s o-sglr for = x the there exst lower trglr trx L R d 9

3 5 pper trglr trx x R U so tht =LU Frtherore V DERVON OF NEW LNER RNSFORON the prevos secto we hve defed eleetry lower trglr trx of order d dex s e We shll fd the bove expresso d the ext theore whose proof fod [8] c be sefl heore he pvot eleets re ozero f d oly f the ledg prcpl sb trces re o-sglr O ths bss the followg c be stted: Corollry Let x R be trx wth o-sglr ledg prcpl sb-trces he there exsts qe dgol trx for = whose etres re: D d re the pvots eleets Now f we scle the trx wth tht dgol trx we wll hve D d oce splfed t c be re-expressed s By sg ths ew trsforto d pplyg the elto process to trx ll of whose etres re tegers ll teredte reslts re tegers too forg ber rg [] sce they re obted throgh ddtos d prodcts hs x x U he prodcts of these reslts tes the fctor re tegers too becse prevosly the step of the elto process sch reslts hd bee ltpled by hs ltplcto process leds to splfcto of the fl reslt We c re-express ths ler trsforto s: e V NEW DECOOSON HEORE HROUGH DEERNNS Usg y of the bove expressos oe c stte ths heore Let x R d he s lower trglr trx ll whose copoets re deterts d U lso hs ll copoets detert for Frtherore roof t follows fro dcto o For = the theore s trvlly tre sce d U For the dcto step we hve: = =

4 5 U = = U he lst row of the trx ltpled for the lst col of the trx s eqvlet to Lplce Expso of tg ot the lst col he we hve : U = -

5 U d he Lplce Expso of sb-trx tg ot the lst col s eqvlet to ltply the -- th row of the trx by the -- th col of he we hve d slr wy we hve Flly hvg the lst row of ltpled by the lst col of we hve 5

6 Now f s the ew trsforto the we hve U order to solve the ler syste of eqtos hve x b x b we Ux b We c se the bcwrd process d solve the ler syste of eqtos sg oly deterts For trx wth flotg pot etres ths process reqres 6 Flotg pot ltplctos V NEW DECOOSON HEORE WH RL VONG Gss elto o rel bers s geerlly stble de to the possblty of fdg rbtrrly sll pvots hs process c be llevted however by exchgg rows drg the elto or cse oly whe pvot s zero we exchge the row slr wy to the Gss process he followg theore s gve wthot proof: heore Let x R Sppose tht the New rsforto row s pertto trces hve bee detered so U tht he the Upper trx U s obted fro wthot exchgg y rows d U Frtherore f exch ber of row s exch exchges we hve V NEW LGEBRC EHOD FOR COUNG HE FFFN pplyg ths ew ler trsforto to sew-syetrc trx R obt the ff of trx U ely: 5 l l l by es of prtl pvotg strtegy we R fro the prcpl dgol of where 6 d 5 roof t follows fro dcto o : = l f d the U Wth ths ew ethod the ff s the reslt of dvdg the lst vle og the frst of the dgol of the trx U: = s the peret slr to detert except tht ech ter the developet s wrtte wth pls sg Next: 5

7 5 U he ff s obted strtg fro process tht cold be descrbed s bcwrd splfcto hs s becse t strts fro the ltte dgol eleet of the trx U d the dvded og the peltte oe d tr og the tepeltte oe d ths wy tl rrvg to the frst vle of the dgol: he ff c be expressed by perets too: O the other hd oe hs tht: U Flly for the geerl cse we hve: he for d

8 55 So: 5 d we c proceeded to obt U he ff s the ltte vle of the dgol of trx U dvded og the eleets of the dgol correspodg to the seqece l= 57 d ltpled by the eleets correspodg to the seqece =6 ely: l l ll V CLCULON OF HE FFFN DUGE RX G WH NOHER NEW LNER RNSFORON he djgte s lso clled the djot We vod ths sge becse fctol lyss t refers to the eqvlet of the cojgte trspose of trx he djgte dj of trx R s the trspose of the co-fctor s trx of the eleets of Coptg the djgte fro ts defto volves the clclto of deterts of order O the other hd the clclto fro the forl dj bres dow whe s sglr d s potetlly stble whe s ll-codtoed wth respect to verso Expressg the ew Ler rsforto ord verso we hve: For d 6 t s sffcet to do the opertos dcted by the ew Ler rsforto x x : to fd the djgte trx of sze : dj Clcltg the trx wth proves hghly effcet whe worg wth the geted trx : dj hs process deds s y s 6 Flotg pot ltplctos other for to express the ew trsforto s 7 where e e e e dj ddtolly wth d prtl pvotg strtegy we hve dj roof Let t sffce to expl how the lgorth wors wth the followg cse Let R d =: dj O the other hd for = we hve: 8 G G G dj dj SOLUON OF X=B WH HE G RX he slteos ler eqtos systes c lso be solved wth ths ew G trx Sce: G dj the:

9 dj x dj b x Gb Gb x hs ew reslt s ff-crer-type solto O EXLE OF SOLUON OF X=B WH HE G RX : VOLERR-LOK EQUON FOR REDOR-REY DYNCS Sppose we hve two copetg speces of fsh lbeled prey sll fsh d predtor lrge fsh Let N d N deote the ber of sll d lrge fsh respectvely he N dfferetl eqtos stsfed by d N the clssc Volterr-Lot odel [] re dn N N dt dn N N N dt = trl growth rte of sll fsh the bsece of lrge fsh = deth rte per ecoter of sll fsh de to lrge fsh = trl deth rte of lrge fshes the bsece of sll fsh = effcecy of trg predted sll fshes de to predto N N = gves the rte of sll fsh beg lost N N = gves the growth rte of the poplto of lrge fsh For speces the syste of dfferetl eqtos s the for dn N jnn j dt j 9 for ll the costts d re j j d postve d re clled eqvlece bers Eqtos re clled the Volterr-Lot eqtos d ply portt role Ecology Sppose we re terested the eqlbr soltos of syste whch cse dn dt Syste becoes N jn j j N or provded tht for = N j j j f the ber of speces s odd = N+ o eqlbr j soltos c exst becse s sew-syetrc trx of odd order Sple d terestg lgebrc cosdertos pert s to stdy d descrbe the stto whe the ber of speces s eve = N Wrtg trx otto we hve x b 5 s eve sew-syetrc trx d x d b re N-desol vectors Eqto 5 s redly solved by the ff-crer s Rle provded tht order N eleets j: \ ff s trglr rry of NN- j 6 N N N N N whch y be expded by ts frst row the se wy s detert except tht the or of eleet j s ff of order N- obted fro by deletg both the -th row d col d the j-th row d col [5] hs f = we hve he Crer s Rle redces ths cse to the rto of two pfffs So we wll clclte the ffs b b b b b b b b b b b b b b b b b b b b b b b Flly x b 7 Otherwse we hve G 56

10 g G f s the -th row vector of G d s the col vector of we hve 8 g b x g X CONCLUSON ths pper we hve trodced ew theore o the decoposto to deterts of trx R d the ew ler trsfortos expressed s eqtos d ost slteos ler eqto systes c lso be solved wth these ew ler trsfortos he reslt s Crer-type soltos O Frtherore we hve obted the ff of expressed s eqto 5 fro the U trx O the other hd we hve proposed odfed Doolttle-Gss-ord elto process two versos: the frst oe ppled to the geted trx d the secod to the geted trx b he frst verso s lgorth to clclte the ew G trx fro R expressed s eqto 8 he secod verso s ew drect ethod to solve ler syste eqto 9 f b R he bove lgorths clclte ff-crer-type soltos of the ler systes Gss elto s slly the ost ecoocl wy to solve x b Nevertheless there s oe reso why ths ew ethod ght be relevt whe R : whle the Gss elto s oreted to trdtol prevlece of the deterts ths ew ethod s focsed to pfff pproch Flly whe referrg to the Crer s rle t hs bee ffred by G Strg [] tht: hs ech copoet of x s rto of two deterts polyol of degree dvded by other polyol of degree hs fct ght hve bee recogzed fro Gss elto bt t ever ws Not oly hs ths fct bee de evdet the preset pper bt lso t hs bee show tht the ff-crer s Rle s ctlly rto of two ffs derved fro the Gss elto process REFERENCES [] r tretse o the heory of Deterts cll d Co Lodo 88 Repr Dover New Yor 96 [] Cyley O the theory of pertts Cbrdge d Dbl thetcl orl Reprted hs Collected thetcl pers 6-6 [] Kth DE Overlppg ffs Electro Cob 996 No rtcle R5 pp rted verso: Cob 996No 7-59 [] RoteG Dvso-Free lgorths for the Detert d the ff: lgebrc d Cobtorl pproches Lectres of the Grdte rogr Copttol Dscrete thetcs Berl 99- [5] Lovász L ler D tchg heory Dscr th Vol 9 North-Holld thetcs Stdes Vol sterd 986 [6] Stebrdge R Notersectg pths pfffs d ple prttos dv th [7] Hoseholder S he theory of trces Nercl lyss Blsdell blshg Copy NY Frst Edto Secod rtg ges: [8] Stewrt GW trodcto to trx Copttos cdec ress c ge: 5 97 [9] Golb GH V Lo Ch F trx Copttos oh Hops Uversty ress ge:56 98 [] GrosswldE opcs fro the heory of Nbers c ll Copy NY ges: [] Strg G Ler lgebr d ts pplctos cdec ress c ges: [] Cello ER Cobtorcs d Reorlzto Qt Feld heory W Bej c Redg sschsetts 97 [] Scott RF d thews GB he heory of Deterts Cbrdge Uversty ress NY d ed pp [] r tretse o the heory of Deterts cll d Copy Ltd Lodo pp [5] Cello ER d Fb S Novo ceto UHOR BOGRHY H E Gozález He receved the hd Opertos Reserch t the Ntol tooos Uversty of exco UN 5 He hs bee worg reserch ctvtes for ore th twety yers He hs pblshed boo d ore th te scetfc ppers resetly he s fll-te resercher t the Ntol sttte of Ncler Reserch NN t the Deprtet of Systes Cro L He receved hs BS Electroc Egeerg fro U resetly he s fll-te resercher t the Ntol sttte of Ncler Reserch NN t the Deprtet of Systes 57