Asynchronous Analogs of Iterative Methods and Their Efficiency for Parallel Computers

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1 Itertol Coferece "Prllel d Dstrbuted Computg Systems" PDCS 23 (Ure, Khrv, Mrch 3-4, 23) sychroous logs of Itertve Methods d her Effcecy for Prllel Computers Msym Svcheo, leder Chemers Isttute for Modelg Eergy Egeerg, 5 Geerl Numov str., Kyv, Ure mvell.u@gml.com,..chemers@gml.com bstrct. hs pper gves the comprso of some tertve methods (Jcob method, Guss Sedel method d the Successve over-relto method) for solvg systems of ler equtos wth ther sychroous mplemettos. Epermets show the effectveess of sychroous tertve methods for solvg ler lgebr problems o prllel multprocessor systems. Keywords Prllel processg, sychroous tertve methods, ler lgebr, effectveess of clculto. Itroducto Comple physcl processes re commoly modelled by usg methods bsed o soluto of smulteous ler lgebrc equtos (SLE). It s dsputble tht soluto of SLE s oe of the mor d prevlg problems clculus mthemtcs. Iterto methods re etesvely ppled usg computg systems, lthough these methods re ot effectve whe pplyg multprocessor computers. Problems mplemetg rse t the sychrozto level of computto process, whe some processors dle t ech terto, wtg for the other processors the system to complete the computto for the curret terto d trsmt the clculted vlues to the other processors. Iterto methods re bsed o tl guess, whch re estmted before the begg of clcultg; durg the clcultg tl guess grdully pproches to the soluto wthout ccumultg errors, whch s especlly urget for lrge SLEs, whe the totl umber of opertos creses drmtclly. fter rechg the prescrbed ccurcy the clcultg stops. s rule, ppromte equlty of vlues t two dcet tertos s crtero for such stop. sychroous terto methods were propostoed to mmze the dles of processors computg system. I [] the theoretcl spects of these methods were vestgted, geerl codtos of sychroous terto process precso were gve d estmto of covergece rte ws obted. However, for ts prctcl mplemetto o prtculr computtol rchtecture wth ether shred or dstrbuted memory t s ecessry to vestgte d specfy the chrcterstcs of the methods d, correspodgly, lgorthms. he preset rtcle dels wth vestgtos crred out for evluto of dfferet sychroous terto methods, otbly the fed pot terto method (Jcob), Guss-Sedel method d successve overrelto method (SOR), comprso wth ther sychroous logs. 2 Orgzto of sychroous terto process ll bove-metoed methods were used for soluto of the system of ler equtos of the type b, () where А s squre mtr of coeffcets the left prt of the smulteous equtos wth of the rght prt of the smulteous equtos, vector of uows., b coeffcet vector -276-

2 Itertol Coferece "Prllel d Dstrbuted Computg Systems" PDCS 23 (Ure, Khrv, Mrch 3-4, 23) Usg computtos, gve ppers [2,3], formul for clculto ws obted: for the Jcob method gve formul (2), for the Guss-Sedel method (3), for the SOR method (4):, b,, (2), b,,, (3),,, ( ) b, (4) where =, 2 ; s the relto prmeter. Let s orgze the followg clculto process multprocessor computg system. Let ech processor smulteously compute ts S compoet group of vector of uows. s ths tes plce, ech processor uses dt, relzed t the momet tme, for geerto of the result of ts group. O completg the clculto of the et group ppromto the processor trsfers clculted vlues to computer memory d they become vlble for other processors. Such pproch s fr both for rchtectures wth shred memory d for cluster computto systems. ltogether, sychroous clcultos c be preseted s epresso [] q ( ) ( q ),, S, (5) where (, S) re curret ppromto of uows vectors, relzed t the momet tme Т; 2 S (,,,,, )', (,,, )' - re curret ppromto of uows vectors, foud t the momet tme Т<Т, tht mr the begg of the et ppromto of the compoet vector of uows group ( ) ( ), dgol mtr, dgol compoets of whch equte to fgure of oe f the correspodet compoet s clcultes by the processor d to fgure of zero f otherwse; q,, S, coeffcets tht equte to the fgure of oe f the tme tervl - s suffcet for the relzto of the vector equle to zero otherwse. ; d ts storge to memory, d Epresso (5) s worthwhle oly f t lest oe tme tervl - s suffcet for relzto of the et ppromto of the vector,.e. t lest oe coeffcet q s equl to zero. Correspodgly, sychroous logs of bove metoed methods orgze coordto wth the followg scheme. t the momet of completo of clculto of the et ppromto of the compoet from the computer memory by y of the processors, ths processor reds out the vector of the curret ppromto of uows, relzed t the momet tme s solved regrd to P, d fter tht we ccept equto regrd to. For the SOR method ( ). P P P (. he the equto f,,,,,,, ) P P P, where Т s the tme of termto of soluto of the I cosderto of peculrtes of the prllel computtol evromets s well s the chrcter of the mtr of the coeffcets of the uows, usg so clled bloc method seems effectve. Let the gve mtr be dvded to submtrces so tht P q q qq, (6) -277-

3 Itertol Coferece "Prllel d Dstrbuted Computg Systems" PDCS 23 (Ure, Khrv, Mrch 3-4, 23) d t s supposed tht ech mtr s vertble. I ths cse bloc Jcob method of the soluto of the system () tht correspods to the troduced decomposto, tes the form, b, (7) where =,, q; =,,, the decomposto of vectors и b coforms to the decomposto А. Hereby, mplemetto of oe bloc Jcob terto requres the soluto q of the systems of the type (7) wth the mtres of coeffcets. For emple, sychroous bloc Guss-Sedel method s relzed the followg wy. he soluto of the ler system () s dstrbuted betwee R processors, ech of whch retreves the uows of ts equto subsystem. t the momet of completo of relzto of the et ppromto of the vector P ( ) by the processor, the compoets of ths vector re wrtte to the shred memory whle the vector of the curret ppromto formed t the momet tme the vector х of the ler system P P,, s red out of t. fter tht the processor strts relzg ew ppromto of P P R P F (( ),,( ),( ),,( ) ). (8) O completg the soluto of ths system we dopt the coveto tht completg of the soluto. ( ), where Т s the tme of Isde the blocs the soluto of systems c be mplemeted sychroously, but tht cse t s ecessry to te to ccout possblty of pperce of stuto whe blocs of uows cot get reewed before the momet of mplemetto of the et terto. he ltter depeds o the level of blce of computtol lodg of processors (whch requres tht ll mtrces hd the sme sze d structure), s well s o the wy of dt trsmsso. 3 he comprso of the clculto tme of terto methods d ther sychroous logs O the groud of formuls (2) (4) were devsed the lgorthms of soluto of the ler system whch terto process must cotue up to the momet whe the vlues become close to the vlues wth requred ccurcy ε. For obtg the epermetl dt we used mtr wth the sze =4 d bsolute morty of ozero elemets, the vector X = {, -,,,, } ws chose s tl guess. I the preset rtcle we vestgted determte logs of sychroous terto methods, where t ech terto were clculted the followg stges: ) 2) 3) 4), 2 covetol mer (bsed o 3, , , {, 2, 3, 4 2 {, 2, 3 2, 4 2 ); re clculted o the groud of X {,,, } ; re clculted o the groud of X } ; re clculted o the groud of X }; 5) etc. Ech method d ther sychroous logs were clculted fve tmes for obtg sttstclly ccurte dt. I the tbles d 2 re gve the obted results for dfferet vlues of requred ccurcy ε, where ter s the umber of tertos ecessry for obtg the requred ccurcy. O the fg. s gve the tme, ecessry for obtg the requred ccurcy for the vestgted terto methods. he vlues ε re gve logrthmc scle. s we c see, usg both sychroous d sychroous mplemetto, the most effcet s Guss-Sedel method. It wll be observed tht successve overrelto method (SOR) the morty of cses proved to be less effcet. he ltter c be epled by the dffculty the choce of relto prmeter ω. Usg Jcob method wth hgh ccurcy we dd t obt results (the vlue of requred ccurcy ε ws t obted)

4 Itertol Coferece "Prllel d Dstrbuted Computg Systems" PDCS 23 (Ure, Khrv, Mrch 3-4, 23) b.. he results of the epermets for the sychroous methods Log (ε) Jcob method Guss-Sedel method SOR method ter, sec. ter, sec. ter, sec ,95 6, ,796 4, ,67 2, ,655, ,779 9,53 9, ,34 7,468 7, ,936 6,328 6,437 -,733 4,249 4,297 b. 2. he results of the epermets for the sychroous methods Log (ε) Jcob method Guss-Sedel method SOR method ter, sec. ter, sec. ter, sec ,859 6, ,834 4, ,62 2, ,54, ,56 8,45 9, ,29 6,392 7,58-2 7,787 5,37 6,473-4,597 4,26 4,38 Fg.. he correspodece of the clculto mplemetto tme d requred ccurcy ε for dfferet terto methods -279-

5 Itertol Coferece "Prllel d Dstrbuted Computg Systems" PDCS 23 (Ure, Khrv, Mrch 3-4, 23) Fg. 2. he correspodece of the problem soluto tme d dmeto of problem wth the sme ε. 4 Cocluso Comprg ech terto method wth ts sychroous log t s obvous tht bsolute morty of cses the ltter provde computtol speedup. s ths tes plce, whe usg sychroous terto methods, cresg the dmeso of problem creses the computto tme reduces. For ech specfc problem t s ecessry to defe whch sychroous log of whch method should be used, whch computtol scheme should be formed d how to orgze the dt trsmsso betwee processors of computtol system. he epermets tht were crred out o qute smple emples show the effcecy of usg the sychroous terto methods for the soluto of ler lgebrc problems o prllel multprocessor computtol systems. he preset results were obted the process of the orgzto of determte computtol process, whe the umber of tertos, computed o oe processor betwee the echge of dt computtol system, s ssged pror. Orgzg completely sychroous process c led to greter computtol speedup. Refereces []. V. N. Beletsyy «Multprocessor d prllel structure to the orgzto of sychroous computto». I Russ Kev: Nu. dum, p. [2]. S.. Belov, N.Yu. Zolotych Lbortory worshop o umercl ler lgebr. I Russ Nzhy Novgorod: Publshg house of the Nzhy Novgorod Stte Uversty med. N.I. Lobchevsy, p. [3]. M.Yu. Svcheo, O..Chemerys "sychroous prllel mplemetto of the method of clcultg the ustedy modes the ler secto of the ppele", Ur - Collected Wors of PIMEE of NSc of Ure. - Kyv, 2. - Vol pp [4]. Gerrd M. Budet sychroous Itertve Methods for Multprocessors // Jourl of the ssocto for Computg Mchery, Vol. 25, No. 2, prl 978, pp [5]. J. M. Orteg, Itroducto to Prllel d Vector Soluto of Ler Systems, Pleum Press,