Modul 2: Solusi Sistem Persamaan Aljabar Linier (SPAL) dengan Metode Eliminasi Gauss dan variannya

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1 Ser temtk Terp tk S o : Sos Sstem Persm Ajr er (SPA) eg etoe Ems Gss vry Sstem Persm Ajr er (SPA) t ke jg seg Persm Ajr er Serempk yk sek jmp m perhtg-perhtg tekk km yg metk sos mers. Beerp metoe sos yg metk sos SPA, try h: sos Sste Persm Ajr No- er (SPAN), sos Persm Dferes Bs (PDB), sos persm Dferes Prs (PDP), Regres er No- er,. Dm mo, pr mhssw S k jk tereh h tk memc g (revew) secr rgks cept tetg eerp pegert sr skr, vektor, mtrks, sstem persm er. Pegg sgt perk meggt yk tr pesert jr S yg sh terp eg mtermter kh mtemtk yg perh kty. D smpg t jg, pr pemc jk tk memhm secr prkts tetg kosep-kosep pemhm m jr er yg mpemetsk m metoe merk. Seteh pegg tetg jr mers, pr pemc jk secr rgks tk memhm kosep-kosep perhtg mers yg erhg eg metoe-metoe Ems Gss Pvot Gss. Kem, eh jh g jk tk mekk ss merk m sos-sos Dekomposs U trks Tr-Dgo. A. Skr, Vektor, trks SPA (). Skr t kostt efsk seg st oyek tgg (ermes o), k g yt (R, re) tp Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (/)

2 Ser temtk Terp tk S g kompeks (C, compe) yg rpy pt kk semrg opers rtmtk/jr secr er, sepert: pemh (s), pegrg (sstrks), perk (mtpks), pemg (vs), perpgkt (ekspoe), s. Dm prkteky, esr skr pt g ts g esr m om g, yt: k R, st skr m g g yt, z C, st skr m g g kompeks. (). Vektor t rg vektor V h st set (sekmp) oyek erp skr (ermes st) yg kepy pt kk opers-opers skr spesfk erp pemh vektor (vector to) perk skr (scr mtpcto). pers-opers terset hrs memeh trtr k, erp: sostf, komttf, strtf. V R, st set skr m g g yt eg jmh ggot seyk h, t V v v v, merpk vektor horzot, segk H [ h h ] h, merpk vektor horsot, Ζ C, st set skr m g g kompeks eg jmh ggot seyk h. (c). trks efsk seg st set vektor yg tersss seemk rp sehgg teetk kmp g eg po perseg-empt, t erorer m (rs) (koom) (ermes ). Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (/)

3 Ser temtk Terp tk S A,,,,,,,,, Dm sstem er, p mmy hy gk mtrksmtrk jr-sgkr sehgg secr seerh: orer mtrks etk eg jmh persm. mg mtrks se tsk m hrf esr (cpt), segk eemeeemey tsk m hrf kec sepert m pes mtrks A ts. (). Sstem persm er, m mo gk sth SPA (Sstem Persm Ajr er) yg efsk seg st set persm-persm jr yg vrevrey erpgkt tgg (er) eg ots erkt:,,,,,,,, ,,,, seemk rp sehgg persm ts pt tsk m ots erkt: t...,,,...,,, [ A ] [ ] [ ] ,,, SPA ts memk h vre t g ( j, j,,, ) yg etk eg jmh persmy. Koefse- Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (/)

4 Ser temtk Terp tk S koefse,j, (,..., ) merpk kostt (keth), emk jg (... ) yg ke seg vektor rs k (VRK). ert koves : eks pertm r eeme,j meytk rs ( ) segk eks ke meytk koom ( j). Agr sos SPA ts pt peroeh, mk persyrt (teorem) erkt hrs peh:. A mempy jw k V tk setp V,. A hy mempy st sos V tk setp V,. Jk A, errt, 4. A - t vers r mtrks A, 5. Determ(A), 6. Rk(A), t mtrks A erorer. Sepert teh jesk ts, mtrks A merpk mtrks jr sgkr. B teorem ts tk terpeh, mk k terj koms er (k megktk persm jr ts ersft SINGUAR). Koms er : per rs, ckp hy rs yg meyeky, per koom, sem rs yg meyeky. B. Sos SPA secr mers Sos SPA secr mers mmy se (hrs) eh efse cept gk eg metoe-metoe ts, sepert metoe Crmer. Nm emk, sos merk secr teks ky jg erke, kre: Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (4/4)

5 Ser temtk Terp tk S eerp persm yg meekt koms er, kt y ro off error r mes peghtg p st thp perhtg y kms ro off error p proses kompts k erkt om g yt (fe pot) m perhtg k termp (overfow), sy kt r jmh persm yg ter esr etoe-metoe sos merk yg yk pk, pt ksfksk seg:. etoe gsg Ems Gss (EGAUSS), prspy: merpk opers ems sstts vre-vrey seemk rp sehgg pt teretk mtrks segtg ts, khry sosy seesk meggk tekk sstts k (ckssttto). etoe Ems Gss secr rgks hs p Prgrf C. Ems Gss-Jor (EGJ), prspy: mrp sek eg metoe EG, m m metoe jmh opers merk yg kk jh eh esr, kre mtrks A megm vers tereh h tk meptk mtrks etts (I). Kre ke terset, mk metoe sgt jrg pk, m sgt ermft tk megversk mtrks. etoe tk hs eh jt m pejr. Dekomposs U (DECU), prspy: mekk ekomposs mtrks A tereh h sehgg pt teretk mtrks-mtrk segtg ts wh, kem secr mh pt mekk sstts k (ckssttto) tk erg vektor VRK (vektor rs k). etoe secr eh jes k hs p Prgrf F, khss tetg metoe-metoe ekomposs U tekk komptsy. Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (5/5)

6 Ser temtk Terp tk S Sos sstem TRIDIAGNA (SDIAG), prspy merpk sos SPA eg etk mtrk pt (st go wh, st go tm, st go ts) p mtrks A. etoe k hs eh jt p Prgrf K.. etoe Tk-gsg (etoe Itertf) etoe Jco, prspy: merpk metoe tertf yg mekk perhr yg peroeh tp ters (mrp metoe sstts errt, sccessve ssttto), etoe Gss-See, prspy: mrp metoe Jco, m metk perhtg mpst, etoe Sccessve ver Reto (SR), prspy: merpk perk secr gsg r etoe Gss- See eg cr meggk fktor rekss (fktor pemoot) p setp thp/proses ters. etoe-metoe tk-gsg sepert ts p my sgt tk efse tme cosmg (memerk CPUtme) yg jh eh esr r metoe gsg. etoe pt ht pejr p k-k merk yg perpstk t toko k. C. Agortm Sos SPA eg etoe Ems Gss gkh #: Ph hrg ) seemk rp yg tk erhrg o. Tetk peg rs seg erkt: () (),,, (, m ;,,, Kem, kostt-kostt peg rs (m) ts gk tk mekk ems term-term p persm-persm smp ke-, sepert erkt: Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (6/6)

7 Ser temtk Terp tk S ( ) ( ) (), j, j m,, j ;,j,, () (), () m ;,, Dm h, rs pertm r mtrks A vektor tk oeh ggg, segk koom pertm r mtrks A () wh go hrs t o. Sstem A () () k tmpk sepert erkt: (), (), (), (), (), (), ( ), () () () Proses ems jtk tk koom-koom, smp ke, gkpk m gkh erkt. gkh k: meggk k tk eks ters, tk retg hrg: k -. B A (k) (k) teh teretk, eg ggp termterm,, K, k teh terems p thp seemy, sehgg mtrks A (k) sekrg memk etk seg erkt: A (k) (), (), (), ( k) k, k ( k), k (), (), ( k) k, ( k), Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (7/7)

8 Ser temtk Terp tk S Ph hrg ) seemk rp yg tk erhrg o. (, Tetk peg rs seg erkt:, k ( k), k ( k) k, k m ; k,, Kem, gk kostt-kostt ts tk megems term-term k p persm-persm k smp ke-, sepert erkt: ( k ) ( k) ( k), j, j m, k k, j ( k ) ( k) ( k) m, k k ;,j,, Brs-rs terh, r smp k, tk oeh ggg, hrg-hrg o mskk p koom k wh go. Deg megkt gkh-gkh sepert ts, seteh gkh ke - k peroeh st mtrks eg ss sepert erkt: (), (), ( ), ( () ) Perhtk, hw mtrks A ts sekrg teh erh mej mtrks segtg ts ( mtrks U), sehgg: U () H errt hw sos hrg-hrg hrs kk m r eks teresr () mej eks terkec (), Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (8/8)

9 Ser temtk Terp tk S yg set seg sstts k (ck ssttto), yg m eg: ( ) ( ), kem, kt eg: k ( k) k, k ( k) k j k ( k) k, j j ; k -,-,, D. Tekk Pvotg m etoe Ems Gss Dm eerp kss, tertm jmp mtrks-mtrk yg ersft sgr kre y koms er, sos secr gsg meggk gortm metoe ems Gss tk memerk hs ketet yg k, hk sergk memerk hs yg meeset jh r yg hrpk. Utk meghr feome terset, perk mofks r gortm ems Gss. P prspy, mofks terset kk eg memperhtk h-h erkt: Hrg pvot m yg teresr r setp rs koom yg ses, yt kompoe, Pemh pvot kk ersrk pemg hrg teresr (mksmm) r setp eeme j, Utk hs terk, seky gk vre press g (DUBE PRECISIN t REA*8). j Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (9/9)

10 Ser temtk Terp tk S E. Cotoh Sos SPA eg Ems Gss Derk SPA erkt: k, sosy h seg erkt: Thp I : Trgrss Ems r persm ke ketg (m h : persm pertm set persm pvot, segk koefse pertm r persm ke set pvot ) Persm pertm kk eg tk megems p persm ke ; Persm pertm kk eg tk megems p persm ketg : Ems r persm ketg (m h : persm ke mej persm pvot, segk koefse r persm ketg set pvot ) Persm ke kk eg tk megems p persm ke : 4 6 Thp II : Sstts Bk (Bck Ssttto) Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (/)

11 Ser temtk Terp tk S Dm r rs ketg (rs terkhr, rs ke ), gsg pt htg hw :, Brs y, m r ers ke - smp rs pertm htg meggk gortm : j,, j j ;,, K, Peroeh : vektor jw yg hsk h [ ] (). stg Progrm Ems Gss (tp Pvotg) C C PRGRA Sos Sstem Persm Ajr er (SPA) t t Persm Ajr er Smt C Dekrs Jes Vre: C IPICIT NNE INTEGER rg PARAETER (rg 7) INTEGER,j,k,eq REA*8 A(rg,rg) REA*8 (rg),(rg) CA system(cer) C Proses Pemsk Hrg Vre: C WRITE(*,) Jmh Persm : READ(*,*) eq D,eq D j,eq WRITE(*,) A(,,,,j,) : READ(*,*) A(,j) WRITE(*,) (,,) : READ(*,*) () C Proses Pemgg Sprogrm Ems Gss-Jor: C CA EGAUSS(eq,A,,) C Pempr/peyj Hs Perhtg: C Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (/)

12 Ser temtk Terp tk S D,eq WRITE(*,4) (,,),() FRAT (X,A,$) FRAT (X,A,I,A,I,A,$) FRAT (5X,A,I,A,$) 4 FRAT (5X,A,I,A,G.7) STP END SUBRUTINE EGAUSS(,A,,) C C SUBPRGRA EIINASI GAUSS: C erpk sos Sstem Persm Ajr er (SPA) eg C formt persm mtrks: [A].[] [], eg rc s C jmh persm jr er (mes SPA) C A mtrks jr sgkr yg ers koefse persm, C vektor vre persm yg k cr hrg-hrgy C vektor rs k yg ers hrg-hrg persm tgg C C Dekrs Vre: C INTEGER REA*8 A(7,7),(),() INTEGER,j,k REA*8 PIVT,UT,TP C Proses sos: () Sstts Ems C D j,- PIVT A(j,j) D j, UT A(,j)/PIVT D k j, A(,k) A(,k) - UT*A(j,k) () () - UT*(j) C Proses sos: () Sstts Bk C () ()/A(,) D -,,- TP () D k, TP TP - A(,k)*(k) () TP/A(,) RETURN END stg progrm (sorce coe) ts msh meggk srote ems Gss yg em kk pvotg p setp pegg eeme. D wh erk srote yg teh meggk tekk pvotg. Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (/)

13 Ser temtk Terp tk S (). stg Progrm Ems Gss (eg Pvotg) SUBRUTINE PGAUSS(,A,,) C C SUBPRGRA EIINASI GAUSS eg TEKNIK PIVTING C C Dekrs Vre: C INTEGER REA*8 A(7,7),(),() INTEGER,j,k REA*8 PIVT,PIVAX,UT,TP INTEGER KPS,pos(7),temp C Proses sos: () Sstts Ems C D, pos() D j,- C ecr PIVT teresr: C PIVAX ABS(A(j,j)) KPS j D j, IF (ABS(A(,j)).GT. PIVAX) THEN PIVAX ABS(A(,j)) KPS ENDIF IF (KPS.NE. j) THEN temp pos(j) pos(j) pos(kps) pos(kps) temp ENDIF C Akhr pecr PIVT teresr: C C wrte(*,*) -pos,pos(j) PIVT A(pos(j),j) D j, temp pos() UT A(temp,j)/PIVT D k j, A(temp,k) A(temp,k) - UT*A(pos(j),k) (temp) (temp) - UT*(pos(j)) C Proses sos: () Sstts Bk C () (pos())/a(pos(),) D -,,- temp pos() TP (temp) D k, TP TP - A(temp,k)*(k) () TP/A(temp,) RETURN END Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (/)

14 Ser temtk Terp tk S Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (4/4) F. Prsp Dekomposs U trks Ietts trks [A] r SPA ekomposs (fktorsss) mej mtrks-mtrk segtg wh () segtg ts (U) seemk rp sehgg ettsy h: [A] [] [U] t A U F. Nots trks U ersrk etoe Dootte Nots mtrks sepert ts tsk s:,,,,,,,,, Perhtk, hw sem eeme go r mtrks ts erhrg (st)! Segk sem eeme ts go semy erhrg (o)! Nots mtrks U tsk s: U,,,,,,,,,,,,,

15 Ser temtk Terp tk S Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (5/5) Perhtk, hw sem eeme yg teretk wh go r mtrks U ts (,, ) erhrg (o)! F. Nots trks U ersrk etoe Crot Nots mtrks sepert ts tsk s:,,,,,,,,,,,,, Perhtk, hw sem eeme go r mtrks ts tk hrs erhrg (st), segk, eeme-eeme ts go semy erhrg (o)! Nots mtrks U tsk s: U,,,,,,,,,, Perhtk, hw sem eeme go (,, ) erhrg (st), segk yg teretk why erhrg (o)!

16 Ser temtk Terp tk S Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (6/6) F. Nots trks A U m SPA Nots trks U seg ekompos mtrks A pt tsk m SPA s: [A] [] [] [U] [] [] Sehgg, m ots etoe Dootte pt tsk:,,,,,,,,,,,,,,,, Segk, m ots etoe Crot pt tsk:,,,,,,,,,,,,,,,, G. Deskrps Thp Strteg Dekomposs Nots A U m etoe Dootte sepert ts pt rk m opers perk mtrks (seg cotoh: mtrks ) s: Brs ( ):,, ;,,,,,,,, K

17 Ser temtk Terp tk S Brs ( ):,,,,,,,,,,,,,,, Brs ( ):,,,,,,,,,,,,,,,,, Brs ( ):,,,,,,,,,,,,,,,,-,,-,,-,,-,- -,-,,,,,,,, # Dr opers-opers perk mtrks U sepert ts, pt smpk eerp h erkt: (). eksme proses ekomposs kk eg cr megs tereh h rs pertm mtrks U. Sejty, megs mtrks p rs tereh tereh (m rs ke-), kem kt pegs mtrks U p rs yg sm, emk setersy smp rs terkhr (ke-). (). Hrg-hrg r sem eeme mtrks U p rs etk eg eeme-eeme mtrks A (mtrks s), Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (7/7)

18 Ser temtk Terp tk S (). Hrg-hrg eeme p koom tk mtrks, pt htg meggk persm erkt:,, /, ;,, (4). Jmh mksmm opers pejmh per eeme mtrks A ses eg jmh/poss rs, (5). P rs reh, gkh/ters pegs mtrks U eh yk gk eg mtrks, seky. Tgs/th: kk h yg sm sepert ts tk kofgrs mtrks U yg ss eg etoe Crot! H. Agortm Dekomposs Kompts Prkts (). Agortm sos merk eg etoe Dootte: Brs : Brs :, ;,,, Pegs mtrks :,,, Pegs mtrks U:,,,,,,,,,4,4,,4,,,, Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (8/8)

19 Ser temtk Terp tk S Brs : Pegs mtrks :,,, (,,, ),, Pegs mtrks U:,,,,,,,4,4,,4,,4 Brs :,,,,,, Pegs mtrks :,,,,,, ( ),,,, ( ),,,,,, ( ),,,,,, Pegs mtrks U:,,,,,,,,, (). Agortm sos merk eg etoe Crot: Seg th, co sr kk ser eg cr megkt gh-gkh tk etoe Dootte sepert ts eg cermt seksm!, Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (9/9)

20 Ser temtk Terp tk S (c). Kompts eg Fortr-77 tk etoe Dootte: C PRGRA Pegj Dekomposs U C Dekrs Jes Vre: C IPICIT NNE INTEGER rg PARAETER (rg 7) INTEGER,j,eq REA*8 A(rg,rg),U(rg,rg) C U(rg,rg) CA system(cer) C Proses Pemsk Hrg Vre: C WRITE(*,) Jmh Persm : READ(*,*) eq D,eq D j,eq WRITE(*,) A(,,,,j,) : READ(*,*) A(,j) C Proses Pemgg Sprogrm Ems Gss: C CA DECU(eq,A,U) C Pempr/peyj Hs Perhtg: C WRITE(*,) trks U yg peroeh: D,eq D j,eq WRITE(*,4) U(,j) WRITE(*,*) FRAT (X,A,$) FRAT (X,A,I,A,I,A,$) FRAT (/,X,A) 4 FRAT (X,F.4,$) C 4 FRAT (X,F.4,X,F.4,X,F.4,X,F.4,X,F.4) STP END SUBRUTINE DECU(,A,U) C C SUBPRGRA DEKPSI U: C erpk sos DEKPSISI trks A mej mtrks-mtrks C U eg formt [A] [].[U] yg hsy smp m U C mes mtrks A (etk eg jmh PA), C A mtrks jr sgkr yg ers koefse persm, C U mtrks jr sgkr tempt peymp hs ekomposs C mtrk A mej U (yg smp sekgs m U). C C Dekrs Vre: C INTEGER,,j,k REA*8 A(7,7),U(7,7),sm Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (/)

21 Ser temtk Terp tk S C Proses pegs mtrks U (m mtrks U): C D j, C Proses pegs mtrks U p rs pertm: C U(,j) A(,j) D, C Proses pegs mtrks : C U(,) A(,)/U(,) sm.d D j,- D k,- sm sm U(,k)*U(k,j) U(,j) (A(,j) - sm)/u(j,j) C Proses pegs mtrks U: C D j, sm.d D k,- sm sm U(,k)*U(k,j) U(,j) A(,j) - sm RETURN END (). Utk Pemhm yg eh mem, coh t progrm m hs Fortr-77 tk etoe Crot! I. ft Dekomposs U tk Sos SPA Sos SPA [A] [] [], me tekk ekomposs mtrks [A], sgt ermft tk meyeesk proem-proem tp moe mtemts yg memetk SPA eg mtrks [A] yg sm tk erg vektor jw, []. Deg tekk ekomposs U, peyees k mej sgt efse yk meghemt wkt p st teh peroeh ekomposs mtrks [A], kre hs ekomposs U terset pt pk tk sem SPA eg mtrks [A] yg etk. Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (/)

22 Ser temtk Terp tk S Betk mm SPA yg meggk mtrks [A] yg etk, sepert setk ts, pt tsk s: [A],,,,,,,,,,,,,,,,,, Perhtk, hw etk ts sesgghy merpk perk etk mtrks, tr mtrks jr sgkr [A] yg ermes eg mtrk seg 4 yg ermes m, eg hs mtrks yg jg ermes m! J. Sos Nmerk SPA me Dekomposs U Sprogrm (SUBRUTINE) wh pt gk tk sos SPA eg etk orm: [A] [] [], meggk mtrks U seg hs ekomposs mtrks [A] eg etoe Dootte. C PRGRA Sos SPA eg Dekomposs U C Dekrs Jes Vre: C IPICIT NNE INTEGER rg PARAETER (rg 7) INTEGER,j,eq REA*8 U(rg,rg),(rg),(rg) CA system(cer) PEN (,FIEptU.t) C Proses Pemsk Hrg Vre r FIE: C READ(,*) eq D,eq READ(,*) (U(,j), j,eq),() C Proses Pemgg Sprogrm Ems Gss: C CA DECU(eq,U) CA SVU(eq,U,,) C Pempr/peyj Hs Perhtg: C Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (/)

23 Ser temtk Terp tk S WRITE(*,) trks U vektor yg peroeh: D,eq D j,eq WRITE(*,4) U(,j) WRITE(*,5) (,,),() CSE() FRAT (X,A,$) FRAT (X,A,I,A,I,A,$) FRAT (/,X,A) 4 FRAT (X,F.4,$) 5 FRAT (5X,H,5X,A,I,A,G.4) STP END INCUDE ecou.s SUBRUTINE SVU(,U,,) C C SUBPRGRA EIINASI GAUSS: C erpk sos Sstem Persm Ajr er (SPA) eg C tekk ekomposs U tk formt persm: [A].[] [], C eg rc s C jmh persm jr er (mes SPA) C U mtrks jr sgkr yg ers koefse persm, C vektor vre persm yg k cr hrg-hrgy C vektor rs k yg ers hrg-hrg persm tgg C C Dekrs Vre: C INTEGER REA*8 U(7,7),(),() INTEGER,j C Proses sos: () Sstts Ems C D, D j,- () () - U(,j)*(j) C Proses sos: () Sstts Bk C () ()/U(,) D -,,- D j, () () - U(,j)*(j) () ()/U(,) RETURN END Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (/)

24 Ser temtk Terp tk S Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (4/4) Tgs! Ujh progrm ts tk SPA erkt: Perhtk eg seksm hs ekompossy (mtrks U) sos vektor -y! K. Sos Nmerk SPA eg trks Tr-Dgo Sos SPA yg eretk mtrks tr-go sergk jmp p proem-proem yg eretk PDP (persm feres prs) yg om secr go (eft postf). K. Betk mm trk Tr-Dgo Secr spesfk, etk SPA yg memk mtrks tr-go pt sjk seg erkt: c c c c t [A] [] []

25 Ser temtk Terp tk S Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (5/5) K. Teorem Sos mtrks Tr-Dgo Jk mtrks jr-sgkr [A] ts merpk mtrks yg om secr go (t eft :postf) memetk mtrks tr-go, mk [A] memk st metk fktorss U yg k, m h k mp U hy memk go: h mtrks wh eg strktr go tm (tsk m mg [ ]) go wh (tsk m mg [ ]); segk mtrks U h mtrks ts yg ers go tm [ ] go ts [c ]. gkh sos yg gk h og eg metoe EIINASI GAUSS. Dm h jk pes SPA ts ssg mej: c c c c Dpt ht eg jes, se ketg go ts mtrks [A] hy s oeh eeme (o), yg errt hw mtrks [A] ts, tk per smp m st vre eretk mtrks, mek ckp hy m h vetor eg pjg msg-msg (mksmm) seesr eeme. Jmh memor tk peymp mej semk sgt errt p st hrg mej sgt esr. H yg per ctt h, hw p setp koom, hy perk h eeme tk-o (k ) yg ems, yg errt jg seg

26 Ser temtk Terp tk S peghemt sh y kompts merk yg retf sgt esr, gk eg peghtg me mtrks peh. Sejty, gkh gortm peyeesy h seg erkt: (). Jk, mk pt ems r persm ke eg meghtg fktor peg erkt: m hsk persm r seg erkt: eg, c m c m (). Deg cr yg sm, jk, pt ems r persm ketg sehgg hsk persm ketg yg r, seg erkt: eg, c 4 m m c Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (6/6)

27 Ser temtk Terp tk S m (c). Tersk cr perhtg ts, sehgg pt smpk p thp-, pt ems r persm (eg sms ) memerk persm r erkt: eg, c m m c m (). Sekes-sekes m tr (c) ts seery merpk sest yg ertr, yt keerg r,,, -, sehgg sstem wy tertrsforms mej mtrks segtg ts (e). Seg sos khry, yt sstts k yg jg mrp eg metoe ems Gss, yt eg megggp hw, k peroeh: kem, tk -, -,,, pt gk: Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (7/7)

28 Ser temtk Terp tk S c stg Progrm trks Tr-Dgo: C C PRGRA Sos Sstem Persm Ajr er (SPA) t t Persm Ajr er Smt eg TEKNIK TRIDIAGNA C Dekrs Jes Vre: C IPICIT NNE INTEGER rg PARAETER (rg 7) INTEGER,eq REA*8 (rg),(rg),c(rg),(rg),(rg) CA system(cer) PEN (,FIEsg.t) C Proses Pemsk Hrg Vre: C READ(,*) eq WRITE(*,*) Jmh Persm :,eq READ(,*) (),c(),() D,eq- READ(,*) (),(),c(),() READ(,*) (eq),(eq),(eq) C Proses Pemgg Sprogrm Ems Gss-Jor: C CA SDIAG(eq,,,c,,) C Pempr/peyj Hs Perhtg: C WRITE(*,*) HASI D,eq WRITE(*,4) (,,),() CSE() FRAT (X,A,I,A,I,A,G5.7) FRAT (5X,A,I,A,G5.7) 4 FRAT (5X,A,I,A,G5.7) STP END SUBRUTINE SDIAG(,,,c,,) C C SUBPRGRA SUSI ATRIKS TRI-DIAGNA eg EIINASI GAUSS C erpk sos Sstem Persm Ajr er (SPA) eg C formt persm mtrks: [A].[] [], eg rc s C jmh persm jr er (mes SPA) C vektor koefse p go wh eg mes -, C vektor koefse p go tm eg mes, C c vektor koefse p go ts eg mes -, C vektor vre persm yg k cr hrg-hrgy C vektor rs k yg ers hrg-hrg persm tgg C Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (8/8)

29 Ser temtk Terp tk S C Dekrs Vre: C INTEGER REA*8 (),(),c(),(),() INTEGER REA*8 PIVT,UT C Proses sos: () Sstts Ems C D,- PIVT () UT ()/PIVT () UT () () - UT*c() () () - UT*() C Proses sos: () Sstts Bk C () ()/() D -,,- () (() - c()*())/() RETURN END. Dftr Pstk Atkso, Ke E., A Itrocto to Nmerc Ayss, Joh Wey & Sos, Toroto, 978. Atkso,.V., Hrey, P.J., A Itrocto to Nmerc ethos wth Psc, Aso-Wesey Pshg Co., Tokyo, 98. Bsmo, Setjo, Kmp Bh Kh etoe Nmerk, Jrs TGP-FTUI, 999. H,.T., S,.C., Comptto ethos Chemc Egeerg, Pretce-H Iterto Ic., Egewoo Cffs, New Jersey, 995. Press, W.H., Fery, B.P., Tekosky, S.A., Vetterg, W.T., Nmerc Recpes, Cmrge Uv. Press, 986. Iteect Property of DR. Ir. Setjo Bsmo, DEA., TGP-FTUI o Sos Sstem Persm Ajr er (SPA) (9/9)