Solving Linear Equations Using a Jacobi Based Time-Variant Adaptive Hybrid Evolutionary Algorithm

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1 Solvg Lear Equatos Usg a Jacob Based Te-Varat Adatve Hbrd Evolutoar Algorth A.R. M. Jalal Udd Jaal Deartet of Matheatcs Khula Uverst of Egeerg & Techolog (KUET) Khula-9203 Bagladesh E-al: jaal@ath.kuet.ac.bd M. M. A. Hashe Deartet of Couter Scece ad Egeerg Khula Uverst of Egeerg & Techolog (KUET) Khula-9203 Bagladesh E-al: hashe@cse.kuet.ac.bd Md. Bazlar Raha Deartet of Matheatcs Khula Uverst of Egeerg & Techolog (KUET) Khula-9203 Bagladesh Ph: Et Abstract Large set of lear equatos esecall for sarse ad structured coeffcet (atr) equatos solutos usg classcal ethods becoe arduous. Ad evolutoar algorths have ostl bee used to solve varous otzato ad learg robles. Recetl hbrdzato of classcal ethods (Jacob ethod ad Gauss-Sedel ethod) wth evolutoar coutato techques have successfull bee aled lear equato solvg. I the both above hbrd evolutoar ethods ufor adatato (UA) techques are used to adat relaato factor. I ths aer a ew Jacob Based Te-Varat Adatve (JBTVA) hbrd evolutoar algorth s roosed. I ths algorth a Te-Varat Adatve (TVA) techque of relaato factor s troduced ag at both rovg the fe local tug ad reducg the dsadvatage of ufor adatato of relaato factors. Ths algorth tegrates the Jacob based SR ethod wth te varat adatve evolutoar algorth. The covergece theores of the roosed algorth are roved theoretcall. Ad the erforace of the roosed algorth s coared wth JBUA hbrd evolutoar algorth ad classcal ethods the eeretal doa. The roosed algorth outerfors both the JBUA hbrd algorth ad classcal ethods ters of covergece seed ad effectveess. KeWords Adatve algorth Evolutoar algorth Tevarat adatato Lear equatos Successve Relaato Mutato. INTRODUCTION Solvg a set of sultaeous lear equatos s a fudaetal roble that occurs dverse alcatos. Lear sste of equatos are assocated wth a robles egeerg ad scece as well as wth alcatos of atheatcs to the socal sceces ad the quattatve stud of busess statstcs ad ecooc robles. Eve the ost colcated stuatos are frequetl aroated b a lear odel as a frst ste. Further the soluto of sste of olear equatos s acheved b a teratve rocedure volvg the soluto of a seres of lear equatos each of the aroatg the olear equatos. Slarl the soluto of ordar dfferetal equatos artal dfferetal equatos ad tegral equatos usg fte dfferece ethod lead to sste of lear or olear equatos. Lear equatos also arse frequetl uercal aalss [2]. After vet of easl accessble couters oe of the a ssue how ca crease the seed to solve equatos. Also t s soete desred to get a rad soluto of the hscal robles for arorate decso. For eale short-ter weather forecast age rocessg sulato to redct aerodacs erforace whch of these alcatos volve the soluto of ver large sets of sultaeous equatos b uercal ethods ad te s a ortat factor for ractcal alcato of the results [3]. If the algorth of solvg equatos ca be leeted effcetl arallel rocessg evroet t ca easl decrease a sgfcace te to get the result. The are so a classcal uercal ethods to solve lear equatos. For large uber of lear equatos esecall for sarse ad structured coeffcet (atrces) equatos teratve ethods are referable as teratve ethod are uaffected b roud off errors to a large etet [4]. The well-kow classcal uercal teratve ethods are the Jacob ethod ad Gauss-Sedel ethod. The rate of covergece as ver slow for both cases ca be accelerated b usg SR techque [2]. But the seed of covergece deeds o relaato factor wth a ecessar codto for the covergece s 0 2 [34]. However t s ofte ver dffcult to estate the otal relaato factor whch s a ke araeter of the SR techque [5]. Awa the Evolutoar Algorths (EA) are stochastc algorths whose search ethods odel soe atural heoea: geetc hertace ad Darwa strfe for survval [678]. Alost all of the works o EA ca be classfed as evolutoar otzato (ether uercal or cobatoral) or evolutoar learg. But Fogel ad Atar [9] used lear equato solvg as test robles for coarg recobato verso oeratos ad Gaussa utato a evolutoar algorth. However the ehaszed ther stud ot o equato solvg but rather o coarg the effectveess of recobato relatve to utato. No coarso wth classcal equatosolvg ethods was gve. Recetl hbrd evolutoar algorths [0] are develoed b tegratg classcal SR techque based o Gauss-Sedel

2 ethod ad based o Jacob ethod to solve equatos whch the relaato factor s self-adated b usg ufor adatato techque. Also obvous bologcal evdece s that a rad chage s observed at earl stages of lfe ad a slow chage s observed at latter stages of lfe all kds of aals/lats. These chages are ore ofte occurred dacall deedg o the stuato eosed to the. B ckg ths eerget atural evdece a secal dac Te-Varat Mutato (TVM) oerator s roosed b Hashe [2] ad Mchalewcz et al. [34 5] global otzato robles. I ths aer a ew hbrd algorth s roosed whch te varat adatve evolutoar coutato techques ad SR techque are used classcal Jacob ethod. The roosed Jacob-based Te Varat Adatve (JBTVA) hbrd algorth does ot requre a user to guess or estate the otal relaato factor. The roosed algorth talzes ufor relaato THE BASIC EQUATION OF JACOBI BASED SR METHODS A lear sste ca be eressed as a atr equato whch each atr or vector eleet belogs to a feld tcall the real uber. A set of lear equatos ukows 2. s gve b the atr-vector equatos: a a2 a b a a a b () a 2 a 22 2 or equvaletl lettg a b atr A ( a ) j ad vector ( ) b ( b ) where s real uber as A b (2) For the soluto of the lear Eq. () Jacob ethod b usg SR techque [2] Eq. () s gve b (k) k k b ajj a j 2 ad k 0 (3) Now coeffcet atr A of the Eq. (2) ca be decoosed as A D L U (4) where D d ) s a dagoal atr L l ) s a lower ( j strctl tragular atr ad U u ).s a uer strctl ( j tragular atr. So atr for Eq.(3) ca be rewrte as: k k H V (5) ( j factors a gve doa ad evolves t b te varat adatato techque stead of ufor adatato techque. The roosed algorth tegrates the Jacobbased SR ethod wth evolutoar coutato techques whch uses talzato recobato utato adatato ad selecto echass. It akes better use of a oulato b elog dfferet equato-solvg strateges for dfferet dvduals the oulato. The these dvduals ca echage forato through recobato ad the error s zed b utato ad selecto echass. Eeretal results show that the roosed Jacob-based te varat adatve hbrd algorth ca solve lear equatos quckl ad effcetl coared to both the classcal ethods ad the Jacob based ufor adatve hbrd ethod. Also ths roosed algorth ca be leeted heretl arallel rocessg evroet effcetl. where H D { L ( ) I U} ad (6) Here - V D b ; H s called Jacob terato atr I s called dett atr ad ( L U) s called relaato factor whch fluece the covergece rate of the both ethods greatl; L ad U are deoted as lower ad uer boudar values of. The otal relaato factor has bee dscussed for soe secal atr [5]. But geeral t s ver dffcult to estate the ror otal relaato factor. THE PROPOSED HYBRID ALGORITHM The ke dea behd the hbrd algorth that cobes the Jacob-based SR ethod wth te varat adatve evolutoar coutato techques s to self-adat the relaato factor used SR techque. For dfferet dvduals a oulato dfferet relaato factors are used to solve equatos. The relaato factors wll be adated based o the ftess of dvduals (.e. based o how well a dvdual solves the equatos). Slar to a other evolutoar algorths the roosed hbrd algorth alwas atas a oulato of aroate soluto to lear equatos. Each soluto s rereseted b a dvdual. The tal oulato s geerated radol for the feld. Dfferet dvduals use dfferet relaato factors. Recobato the hbrd algorth volves all dvduals a oulato. If the oulato sze s N the the recobato wll have N arets ad geerates N offsrg through lear cobato. Mutato s acheved b erforg oe terato of Jacob ethod usg SR techque as gve b Eq. (5). The utato s stochastc sce used the terato s tall geerated betwee L (=0) ad U (=2) ad s adated stochastcall each geerato (terato) ad adatato ature of s also te varat. The ftess of a dvdual s evaluated based o the error of a aroate soluto. For eale gve a

3 aroate soluto (.e. a dvdual) z ts error s defed b e(z) = Az b. The relaato factors are adated after each geerato deedg o how well a dvdual erfors ( ter of error). The a stes of the Jacob-based hbrd evolutoar algorth descrbed as follows [0]: Ste : Italzato Geerate radol fro a tal oulato of aroate solutos to the lear Eq.() usg dfferet relaato factor for each dvdual of the oulato. Deote the tal oulato as (0) (0) (0) (0) X { 2 N } where N s the oulato sze. Let k 0 where k s the geerato couter. Ad talze corresodg relaato factor as: d L for 2 d for N U L where d N (7) Ste 2: Recobato k c ( kc) ( kc) ( kc) Now geerate X { 2 N } as a teredate oulato through the followg recobato: ( k X c) R X ( k) t (8) Where R ( rj ) NN s a stochastc atr [6] ad the suerscrt t deotes trasose. Ste 3: Mutato k The geerate the et teredate oulato X k c fro X as follows: For each dvdual kc k c ( N) oulato X roduces a offsrg accordg to Eq. (5) ( k) ( kc) H N V (9) Where s deoted as relaato factor of the th dvdual ad s deoted as th (utated) offsrg so that ol oe terato s carred out for each utato. Ste 4: Adatato k k Let ad be two offsrg dvduals wth relaato factors ad ad wth errors (ftess value) e( ) ad e( ) resectvel. The the relaato factors ad are adated as follows: (a) If e( ) e( ) () the ove toward b usg ( 0.5 )( ) (0) ad () ove ( U L awa fro ) ( ) whe whe usg () E N T Where E N(00.25) T ad are deoted as Te- Varat Adatve (TVA) robablt araeter of ad resectvel. Here T l( ) t 0 (2) Whch s the Basc Te-Varat (BTV) araeter whch λ s a eogeous araeter used for creased or decreased of rate of chage of curvature wth resect to uber of teratos t. Also N (0 0.25) s the Gaussa dstrbuto wth ea 0 ad stadard devato Now E ad E deote the aroate tal boudar of the varato of TVA araeters of.e. (-E E ) ad resectvel. Ad factors corresod to.e. (-E E ) & ad are adated relaato. (b) If e( ) e( ) the adat ad the sae wa as above but reverse the order of ad. (c) If e( ) e( ) o adatato. So that = ad. Ste 5: Selecto ad Reroducto Select the best N/2 offsrg dvduals accordg to ther ftess values (errors). The reroduce of the above selected offsrg (.e. each arets dvdual geerates two offsrg). The for the et geerato of N dvduals. Ste 6: Terato If { e(z) : zx} < (Threshold error) the sto the algorth ad get uque soluto. If { e(z) : zx} the sto the algorth but fal to get a soluto. Otherwse go to Ste 2. THEOREMS The followg theore establshes the rad covergece of the hbrd algorths. Theore-: If there est a 0 such that for the or of H H the l ( k) k * where s the soluto vector to the sste of lear equatos.

4 Proof: The roof of ths theore sle ad straghtforward ad roof of ths theore s gve [0 ]. The followg theore justfes the adatato techque for relaato factors used roosed hbrd evolutoar algorths. Theore 2: Let ρ( ) be the sectral radus of atr H be the otal relaato factor ad let ad are the relaato factors of the selected ar dvduals ad resectvel. Assue ρ( ) s ootoc decreasg whe ootoc creasg whe ρ( ) ρ( ). The ad ρ( ) s. Also cosder () ρ( ) ρ( ) whe (0.5 )( ) where E E ] ad [ () There s a ver hgh robablt that ρ( ) s less tha ρ( ).e ρ( ) < ρ( ) whe ( ( [0E ]. U L ) ) whe whe where Proof: The frst result ca be derved drectl fro the ootoct of (). The secod result ca also be derved fro the ootoct of () wth a ver hgh robablt as []. PERFORMANCE OF THE HYBRID ALGORITHM I order to evaluate the effectveess of the roosed JBTVA hbrd algorth uercal eerets have bee carred out o a uber of robles to solve the sstes of lear Eq. () of the for: A b The followg settgs are vald all through the eerets: The deso of ukow varable s 00 oulato sze N =2 boudar of relaato factors ( L U) = (02) (.e. ol two dvduals were used so that tal s becoe 0.5 ad.5 resectvel) the aroate tal boudar E ad E are set at 0.25 ad resectvel the eogeous araeter λ s set at 50 each dvdual of oulato X s talzed fro the doa 00 (-30 30) radol ad uforl ad the stochastc atr R was chose as follows: If the ftess of the frst dvduals was better the the secod let (3) else let ( k c) ( k c) ( k c) ( k c) ) Each eeret s ru 0 tes usg 0 dfferet sale aths ad the averaged the. Now the frst roble s to solve lear equatos Eq. () where a = (-7070); a j = (07); b = (070) j (.e. roble P 2 Table I). A sgle set of araeters are geerated radol fro the above etoed roble ad the followg eerets are carred out. The roble s to be solved wth a error 6 saller tha 0 (threshold error). Fg. shows the uercal results ( grahcal for) acheved b the roosed classcal Jacob based SR ethod wth several relaato factors ( = ad.5) ad roosed JBTVA hbrd algorth. Ad Fg. 2 shows the uercal results ( grahcal for) acheved b the roosed classcal Gauss-Sedel based SR ethod wth several relaato factors ( = ad.5) ad roosed hbrd algorth. It s observed Fgure ad Fgure 2 that the rate of covergece of JBTVA algorth s better tha that of both classcal Jacob based SR ethod ad Gauss-Sedel based SR ethod ecet for = 0.5 where Gauss-Sedel based SR ethod coverges a bt fast tha JBTVA ethod. It s also observed that both classcal ethods are sestve to the relaato factors whereas JBTVA algorth s ot so. Fgure 3 also shows the uercal results ( grahcal for) acheved b the roosed hbrd algorth ad JBUA (Jacob based Ufor Adatato) [] hbrd algorth. It s observed Fgure 3 that the rate of covergece of TVA-based algorth s better tha that of UA-based algorth. Table I resets te test robles labeled fro P to P 0 wth deso 00. For each test roble P : = the coeffcet atr A ad costat vector b are all geerated uforl ad radol wth gve doas (show 2d colu wth corresodg rows of Table I. Ths table shows the coarso of the uber of geerato (terato) of the JBUA ad roosed JBTVA hbrd algorths to the gve recseess (see colu three of the Table I). Oe observato ca be ade edatel fro ths table ecet for roble P 0 where the JBUA algorth erfored ear to sae as JBTVA algorth TVA-based hbrd algorth erfored uch better tha the UAbased hbrd algorth for all other robles. 2

5 Fgure 4 shows the ature of self-adatato of the UA-based hbrd algorth ad Fgure 5 shows the ature of self-adatato of the TVA-based hbrd algorth. It s observed Fgure 4 ad Fgure 5 that the self-adatato rocess of relaato factors TVAbased hbrd algorth s uch better tha that of UAbased hbrd algorth. Fg. 5 shows that how tal = 0.5 s adated to ts ear otu value ad reaches to a better osto for whch rate of covergece s accelerated. O the other had Fgure 4 shows that tall = 0.5 b self-adatato rocess does ot graduall reaches to a better osto. PARALLEL PROCESSING The arallel searchg s oe of the a roertes of evolutoar coutatoal techques. Now sce classcal Jacob based SR ethod ca be leeted arallel rocessg evroet [23]. So JBTVA as lke as JBUA [] ca also be leeted heretl arallel rocessg evroet effcetl. Where as Gauss-Sedel based hbrd algorth [0] heretl ca ot be leeted arallel rocessg evroet effcetl. CONCLUDING REMARKS I ths aer a Te-varat adatve (TVA)-based hbrd evolutoar algorth has bee roosed for solvg sstes of lear equatos. The TVA-based hbrd algorth tegrates the classcal Jacob based SR ethod wth evolutoar coutato techques. The te-varat based adatato s troduced for adatato of relaato factors whch akes the algorth ore atural ad accelerates ts rate of covergece. The recobato oerator the algorth ed two arets b a kd of averagg whch s slar to the teredate recobato ofte used evoluto strateges [67]. The utato oerator s equvalet to oe terato the Jacob based SR ethod. The utato s stochastc ad te varat sce the relaato factor s adated stochastcall. The roosed TVA-based relaato factor adatato techque acts as a local fe tuer ad hels to escae fro the dsadvatage of ufor adatato. The effectveess of ths hbrd algorth s coared wth that of classcal Jacob based SR ethod ad Gauss- Sedel based SR ethod. Also uercal eerets wth varous test robles have show that the roosed JBTVA hbrd algorth erfors better tha the JBUA hbrd algorth. Ths relar vestgato has showed that ths algorth outerfors JBUA hbrd algorth as well as classcal SR ethods. Jacob-based hbrd algorth s also ver sle ad eas to leet both sequetal ad arallel coutg evroet. REFERENCES [] A. Gourd ad M. Bouahrat Aled Nuercal Methods Pretce Hall of Ida New Delh (996). [2] L. A. Hagaa ad D. M. Youg Aled Iteratve Methods New York. Acadec ress (98). [3] Krshaurth E. V. ad S. K. Se Nuercal Algorths coutatos Scece ad Egeerg Afflated East-West Press New Delh (989). [4] F. Curts Gerad Patrck ad O. Wheatle Aled Nuercal Aalss 5 th edto Addso-Wesle New York (998). [5] D.Youg Iteratve Soluto for Large Lear Sste New York Acadec (97). [6] M. Schoeauer ad Z. Mchalewcz Evolutoar Coutato Cotrol ad Cberetcs Vol. 26 No (997). [7] T. Bäck ad H. P Schwefel A overvew of Evolutoar Algorths for Paraeter Otzato IEE Tras. O Evolutoar Coutato Vol. No (993). [8] T. Bäck U. Hael ad H. -P. Schwefel Evolutoar Coutato: Coets o the Hstor ad Curret State IEEE Tras. O Evolutoar Coutato Vol. No.. 3-7(997). [9] D. B. Fogel ad J. W. Atar Coarg Geetc Oerators wth Gaussa Mutatos Sulated Evolutoar Process Usg Lear Sstes Bol. Cberetcs Vol. 63 No-2.-4 (990). [0] Ju He J. Xu ad X. Yao Solvg Equatos b Hbrd Evolutoar Coutato Techques Trasactos o Evolutoar Coutato Vol.4 No (2000). [] Jaal A R M J U M. M. A. Hashe ad M. B. Raha A Aroach to Solve Lear Equatos Usg a Jacob-Based Evolutoar Algorth Proceedg of the ICEECE Deceber Dhaka Bagladesh (2003). [2] M. M. A. Hashe Global Otzato Through a New Class of Evolutoar Algorth Ph.D. dssertato Saga Uverst Jaa (999). [3] Z. Mchalewcz Evolutoar Coutato Techque for Nolear Prograg Proble Iteratoal Tras. o Oerato Research Vol. No (994). [4] Z. Mchalewcz ad N. F. Atta Evolutoar Otzato of Costraed Probles Proceedg of the 3 rd Aual Coferece o Evolutoar Prograg Rver Edge N3 World Scetfc; (996). [5] R. Saloo ad J. L. Va Hee Acceleratg Back Proagato Through Dac Selfadatato Neural Networks Vol. 9 No (996). [6] E. Kreszg Advaced Egeerg Matheatcs 7 th edto Joh Whe & Sos New York (993).