A node ordering algorithm to speed up the solution of sparse matrix and sparse vector linear equation systems

A NODE ORDERING ALGORITHM TO SPEED UP THE SOLUTION OF SPARSE MATRIX AND SPARSE VECTOR LINEAR EQUATION SYSTEMS An onio G6mez and Leopoldo G. F anquelo Dep o. de Ingen. Elec ica, Elec 6nica y Au oma ica Uni e sidad de Se illa, Spain ABSTRACT. Recen ly, mo e a e n ion hag be en do o ed o spa ge ec o mo hoda in o d . ~ o e duc e he compu a ion a l bu den when aol in~ spa ae sys ems o! linea equa io n s. Theae me hode exploi he spa si y o he independen ec o and/o he deai e o know only a subse o ho unknown ec o . They a e al a o a pplic a ble when e a c o iza ion o • s ll a h ly modi i ed ma ix i . . qui e d . This pape p opOS8a a scheme o o de he nodes wi h he pu poae o educing he numbe o ope a ions when applying spa aB ec o me hoda. INTRODUCTION. Spa ae ma ix equa lonu o he o m: Ax • b ( 1 ) a ppea 1n ma ny e n~ine e in~ i e ld s . Tho s anda d o olu ion p ocod u e pe o ms a ap a si y - o ian ed LOU decomposi ion o A. ollowed by !o wa d a nd ba ck ope a iona On ho independen ec o , b. Thl . Pa pe d .a l . wi h wo kind . o p oblemsl a) Repe a ed solu ion a (1) when ma ix A 1s sligh ly modi!ied (pa i a l ma ix . a c o i:e ion p oblem) Ill. b) Solu ion o (1) when ei he VQc o b ha D only a ew non - ze o elemen s o a small numbe o elemen s in he unknown ec o x i8 needod (8p a ~ e ec o p oblem) [2l. lIS compeno~llon ~elhod •• he~e •• pecla a e 8l.on~ly ala ad o •• l op. ~ lon con ol o . acl.leal powa .ya .ma. A ecan pape by Tinney. .1. (2) emph •• ized he in a ea o apa •• ~c o ~. hod. 1n powe .y. .~ analyala, .hQwln~ a d ama ic .4u~ lon 1n opa a lona CQun~ co~p. .d o con enllonal Me hod., In lh~l pape , lh. ~ulho . noled lh~l naw be de elop~d 1n o de o enhanca he nod. o de lnl alco l hma had o apa al y o L- 1 <U- 1) wi hou In his p.ape , p .,) lJ, .. .1 U)' Tinney' •• chama J (T-l) o da 1ne •• lao known •• mlnlmu" de~ ~e .l~o llh~ [)) . BASIC CONCEPTS. Fo almpl i ci y h _ p. ~ . will b. •• ic ed o .~·. .. eymmel le ma i c es o ma .lea. ha a e aymm. le In pa la n o non ze o Gi en .. aymm8 ic ma ix A-<alJ) an ~'8oclaled undi ec ed , aph G=( V,£ ) can ba da ined. , n,j 1. ~h. e V ie he I o( e ice. (Vl, 2'" . nl ( uno de ed pai . o elamen e o V (ad.ea), also called noda., poin ~, bu •••• e e, and o he namee o adl.' al"'l b oln c hes, ol CI o linea, I (Vi' J)E . £, han Vi an c. J a e .aid o ba .,jJ.c~n . Tho ,j., ol o a e ex 1e he numba o ~ lcaa adjacen o i . 11 C h a ~ n e ica •• han. one- o-ona map. . om V on o he .a 11.2, __ . ,n) 18 called a numbe lnc o C, d ced & aph •• ~ocl~ ad wilh lhe educed ma ix I" ob ained ( om he o lg1noll , olph by de ~ e ln, e ex and inciden a ' CI and adding a ca J 11 ba w •• n any pal o e ic •• which . . n.l~hbo . o nel,hbo . o .ach olhe . 1 bu . . no . Cone1d., he .ym~. lc laclo lxa lon o • apa ••• ~. lx A in o UlU. T ,g ".ph a •• ocla ed o he m. l~ _u +u ,_ called he illed ,".aph G - .ulu lun oC (1), 10111 .... ollli. b 111 apa •• 'U' only .. .w _lam_nle cd " . . need.d. ollowing (21 .o~e de lnl lona al. ed o .pe ce ec o . will be in oduced In o de o make he papa .el -Mu lclen . A alncl. on 1_. ec o wi h only on. nonze o ele~~n . ( •• ~. 1n loca ion ~). A pa h (o a single on ,_ de ined .8 an o de ed I & oC OWIl o U which . e a ic ly neeea.a y {o he lo wa d solu .lon o olgmen o X Is wan ed. Such. pa h 1. • •• ily de e mined om he ape ae a uc u . o U as ollowa (21 : 1) Le k be he i a ow 1n he pa h . 2) G. he numbe o he 10w. A -nu~be ed nonze o ele~en ! n ow l:. o U. Replacs k wi h his numbe and include i in he pa h. ) I k 1. he laa ow, .x1 . O he wis., e u n o s ep 2. Uhan ow k o A 1. modi ied, g aph a he single on k- h ~ua only .h. ow. belon&:i ., b. aken in o accoun o he pa h. du in&" he . ac o iza ion o A. Hance, pa ial ~a ix e ac o 1za ion and o wa d .olu lon p oca •• a a .bo dad 1n he •• ma way. P o. .o~ on; only ha (o ola d and backwa d p occ ..... ' on he ind.pendon ec o will b. cona lde ed. I only ha owa o a ¥l en pa h a e in ol ed ln he solu ion o (1) he co esponding p oce8S la called he Faa Fo wa d llupecL l ll ly. (F ) o A ; iln example conulde he Iii node IEEE es &ya e:n (41, who •• Ina lx U il uhown 1n Fl~ u o 1 .• . The o da in£, al c. 1 .hDl uae(l i ! T-2, JI9 and node ( he a e ence noda In he Load Flow p ob18~) 1. dla eaa dod. A pa h c aph o hla ne wo k. which compac ly de.c ib •• • 11 o i •• 1n~le on pa h. I .. ahown In Fleu . l.b. The ~ .ph 1 ... •• which h .... di ec ion •• n •• ( om low o hleh nod. o and om hLeh o low nod. (o S. The a 1_ a ala ion.hlp be ween his pa h aph and he apa al y s uc u e o U-1, The nodee belon !n o he k- h noda pa h coincide wi h he nonzo o columma o he k- h ow o U- 1• H. lx U- 1 (o ho o mQ example 1_ i an In FI,u . l.c. A. may ba no ad, he numbe o nonze o .lamen e o U- 1 1. di ec ly el. ed o he a e .c •• 1n~1. on pa h l.na h. Al~o he nu~b. o nonze o .lemen . o U 10 di ec ly ela ed o he . e .~e mul -adds needed o pe o m he o wa d o back aUba i u ion p oce.s on .ach UESCRIPTION OF THE ALCORlnU1. The basic idea which mo i . ed he alzo i hm is qui e sImple. La Ua obee e &&81n he T-2 o de inc applied o ha l~-node ey. e~ ahown In i~u e 1. In hi. ca •• ho a . a~e pa h l.n~ h o he •• 1. 4.92~ Thi. hi~h alue 1e due ~alnly o he exi. ance o a lon~ b anch consis ing o e node. (2,3,4,6,9,11,12,13). A88ume now h. ha s.me ays am 1. eo de ed •• can ba aeen In icu e 2. Tha esul an & o az. pa h l.n~ h Is 3. 61 and he lonse. b anch la compo8od only a $ nodo •• Th18 be e •• ul could be .xpec ed by _ .1~ple ins pec ion o he ee, whe e wo ae . o nod.s a e .pp~ en (enci clo d ) who.e p.lh. ha e only wo common node •• Thl. shapa haa b.an achie ed by uainc 8 clus a in¥ slco llhm in o de o ob ain wo waakly in e connec ed clue e •• Th. educ ion in he a e a,a pa h leng h impli.. a ~o e epa •• U- I ma ix . I may ba obee ad, howe e , ha h . 1. done a u.o coa o a ell,h ly la ,a Ill-in o U (one ex a ele~en appea s). IN Honce, moll ~ ed by he abo e eauI e, he p ocedu e 18 p opoaadl ollowlnc heu is ic .) Pe o m a clus e ine o he ma ix A, which yields he ypical Bo de ed Block 018£00_1 Fo m (BDO ) . The op imum numba o elua e a 1s no ~nown -. p lo i". In .ana e1, 80me l i.l. mua be ca ied ou , becau.a a la C8 nu~b. o elua a.. may deg ade auba an ally lhe spa .1 y o U, o e ldlnc lh. ,.1n e ained wi h. aho a pa h I.oa h . So~. l~.a. he ~. lx a uc u e I ,.l aUle •• a he numbe o elua a a o be adop ed. b} Reo de lhe node. belon,!n, o .ach elua e ollow!n, he minimum dec ee .l,o llh~ (T-2). A ew clus e ing algo i hms ha e appea ed In he lI e a u e (a.e lo ln~ .nco (5.6.7]). All o h.~ ha . he d awback o •• lou.~y deE .dln, he .pa . y o U, a. h_ .1~_ o he bo de c ows up . Recen ly a clus e inc .l&o l h~ ha. b •• n p oposed (Bl which has p o ed o ~ old h1s p oblem almoe comple ely . Thl. 1. he on. adop ed in he ne ~l sec ion. DISCUSSION O EXPERIKENTAL RESULTS. Th. adml .nc. ~a lx a ~any elec ical powe eye lme hee b.en used o e s he p oposed alco l hm. The esul e co espondin& o he 81Eh la ge sys ems a e abula ed n Tebles I and II o he T-2 al.o l hm and he one p opo.ed he e •• pec l .ly. The " Appendix d •• c ib •• b ie ly h. che ac e l. lc. a s1ze. anE. om 116 o 661 nod ••• a. ~ell •• he ac o iza ion o he ~. lx (ac ually. he ~e lx has one nod. le •• h.n he nu~b. lndlc. ed In he l a column ••• he e . ence o .lack node Ie no included). Fo e e y po •• ble .Incl. on he pa h lenc h end he o .l I 1/ i II II I I I I , ! I1UL -1 ADD HOD U U Ae 118 25) •• 0 425 175 399 2761 710 265 5<9 3622 912 29) 68' 5435 1229 )83 1038 8971 2503 ,,8 1189 10667 2~50 596 1710 15688 007 ~hl 18~1 17638 <4972 uL -1 ADO HOD U U Ae 118 256 968 436 175 <l3 1961 B8 265 552 30426 .87 293 121 396] 1414 38) 1048 T012 1602 "8 1197 8-41b ;1916 5'6 1714 1]819 055 661 1855 lS 3H 5;:'10 RASC O" SlNCLEi OX S CLU S TE R ED SI~CLETO~S I SINGL 2 SHICl .5 SING 10 SINC 2 SIHCL 5 SIHCl 10 SINC A? 1.01 loP AD AP AD A? AD AP AD AJ' AD AP AD '.5 21 1·4. 1 )) 25.1 62 35 . 9 88 10 . 6 2) 1] . 1 2' 17.5 ,0 16.9 50 22.6 66 33.2 9. 0.3 130 19.0 56 19.8 57 H . 2 68 1~.1 .. 21.3 : 10 .36.2 111 51.5 148 16 . 1 53 18.1 56 23.8 70 19.6 57 28.5 85 "".1 129 60.~ 175 20.1 56 23.6 68 29.8 ., 2~.5 137 31.~ 1 81 ~9.7 260 66.1 32<4 26.3 H5 28.2 H9 H . 6 173 2~.8 1.041 H.9 189 50.9 268 70 . 7 lH 25.8 14] 28.] 152 34.3172 27. 04 188 37 . 1 257 58.9 392 53.5 512 28.6 194 30.9 203 36.6 220 21. 7 190 40.3 278 61.8406 86.0 523 27.8 187 32 . 1 210 37 . 3 221 I Table I. Rc.ul . ob ained wi h T-2 ~. hod . RANOOM SINGlETCNS ClDSTERED SINGLETONS 1 SINGL 2 SIHGL 5 SHiel 10 S NG 2 S HC 5 SINGL- o SIHC AP AD AP AD AP AD AP AD AP AD AP AD AP 1.0 .. ) 21 13.8 13 21 . 5 59 l5.~ 89 10.6 ' 14 12.2 27 11.7 <l 12.) 13 18.3 54 32 .0 .8 46 . 1 138 13.8 . 37 16.1 <l 10.8 56 14.0 <S 20.4 65 304 .9 110 50.8 149 1-4 . 6 ,. 17.5 53 22 . 7 65 13.6 ,2 21.8 72 37.9 130 58.1 196 15.2 ,7 16.8 ' 51 2l.9 6 19.4 .1 28.1 147 45 . 9 231 66,8 328 20.9 104 23.1 III 29.1 134 19.8 101 18.0 146 H . 9 24b 70.1 J4" 21.9 ill 23.6 11-4 29.6 IH 14 . 2 153 36 . 3 245 58 . 7 392 81 . 9510 25.3 162 18.7119 3~.9 102 24.2 IS6 36.6 20 60.739J 84.8 519 2(.8 155 1 9. 6 lB 35. 4 203 A~: A e .,e pi h l~n' h. A O: A e a,e o al ~ul -.dd • . Tabl II. R .ul . ob alned wl h he p o pol d .l, o l ha. 01 R2 OJ .. 5< 12 8 H 56 H 12 II 5< 71 • 23 5< 72 8 26 57 11 13 28 56 70 12 25 56 68 11 23 55 68 10 21 ~ Rl .2 0) .. ~ I 54 H 8 25 9 ! S4 13 8 28 . , 5-4 H 8 27 2 53 72 6 18 , 55 13 9 H 3 54 H 8 23 " 55 H 9 26 J 5~ J4 8 2' J ! J22 abula ed, •• well a. he a lo. Rl. R2. A3. and R4 de ined 1n (21 which el e & ~e •• u . o he el. i e ad en _,e o uelo& lne .ad o he lull o wa d p oc •••• u he e. . wa . ~.d. ln ol !nc epa a. ec o a wi h wo, i e end len .ndo~ly cho •• n nonze o al.men a. o .ach c... 100 lal. wa e pe (o ~ed. and he a a ac_ pa h lenc h a. wall.. he a e a,e lo al mul -add. 1n a e abul. ed he e. F om h... •• ul . lh. a ios R5 and R6 de ined 1n (21 may alao be compu ed. Th ••• a loa indica o he loa. o . iclency o epa •• ec o me hode a. he numbe ~( andom nonze 08 c owe. In p ac ice, he nonze 08 1n he epa .. eclo a e no andomlY chosen. To ea he . ec . o opolo11cally el. ed nod •• ano he .a o( 100 ial. was done. Thi. 1me he wo. i e anJ en nonze o l~~onl& o ll ~ spa ae eclo we e cho •• n 1n a .1mil. way aa la done In dl~&onal b~nd o do ing. A nod a i. andomly chossn •• he i s one. The adjacen nodes o hose al eady numbe ed a e conaecu l ely choaen. and so on unlil he equi ed numb. o nodas is achie od. The a e age eGul s om lhe~. .ale a . aleo el en . I can be seen, •• poin ed ou 1n 121. ha he g ow h 1n pa h he nonzo o enl l.~ The las column in Table II ahowe he adop ed numbe o clus o s o ~ach n~lwo k. These esul s sugg.e he ollowin, conclueionsl The (ill-in o( lh. ma ix U is sligh ly inc .aead wi h •• epec o T-2 al~o i hm. as was expec ed (abou l~). OpPosilely, he spa si y o 0- 1 La ai&ni(ican ly imp o ed (abou IS~) which means. a. wa. axplained p e iously, ha he a e .ce pa h ~~n h o ana 51n la on ia p opo ionally dac s •• ad (column 5). Beslde&. he a e age o al mul -adds in FF (o 1 81n~le on (column b) 1s abou 20~ lecs han wi h T-2 algo ilhm. , ,. J23 As he numbe o sin~le on. ~ ows, bo h a l~o l hma end o behA e simila ly, hou~h (o 10 clus e e d .in~le ong he p opoaed & l~o i hm s 111 sa es abou 1~% o mul -adds 1n co~p a .d o T-2 (colu~ 18) • -When h~ aingle ona a s andomly chosen, he ad an ~ge o( he p oposed algo i hm o e T-2 is las8 impo an . CONCLUSIONS. Recen ly, mo e in e es hes boon de o ed o op4C'Qe VQc o me hode, · whlch enhance o wa d and backwa d a ubs 1 u 10n ~o=e==o= · b7 exploi ing he spa s1 y o he independen ec o and/o ho need o know only a subse o hs unknown ec o . The sa no ochnlque a con be usod In h e pa ial ma ix e a c o iz a ion p oco ~Q . Do h ypes o p oblems a ise equen ly In many eng1nee ing ields, pa icula ly in elec ic powe sys ems analysis. The speedup in a pa icula applica 10n dep en ds no only on ho numbo o nonze oo in he oc o bu on he epa el y o U and U- 1, which may be enhanced wi h a p ope node o de ing. In his pape , an a lgo i hm is p opoaed which clea ly imp o es he spa si y o U- 1 compa ed o he minimum de~ ee alg o i hm. The imp o emen 10 anola ed in o a educ 10n on he ope a iona coun bee ides 15:1:. REFERENCES. [1] - Chan S.M., B andwajn V., Pa ial ma ix e ac o iza ion. IEEE T ansac. on PWRS-1, pp. 193-200, 1986. [2] -Tinney W.F., B andwajn V., Chan S.M., Spa se Vec o Me hods. IEEE T ans. on PAS-104, pp. 29S-301, 1985. [3] -Tinney W.F., Walke J.W., Di ec Solu ions o Equa ions by Op imally O de ed T iangula P ocee. IEEE ol. 55 pp. 1801-1809, 1967 Spa se Ne wo k Fac o iza ion. [4] - IEEE Commi ee Repo , IEEE Reliabili y Tes s Sys ems. IEEE T ansac. on PAS-98, pp. 2047-2054, 1979.