MATRIKS & SOLUSI PERSAMAAN LINEAR

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1 MATRIKS & SOLUSI PERSAMAAN LINEAR Pd bb dbhs kosep dsr d metode d dlm meyelesk persm ler deg pedekt mtrks terutm berkt deg ksus-ksus khusus dlm fsk. Dsjk beberp metode komputs umerk, melput metode elms Guss deg pvotg, metode Guss-Sedel, d mtrks Trdgol yg cukup fmlr d terpk dlm mslh l ege dlm fsk kutum, sebg stmul utuk pemhm yg lebh tesf terhdp metodemetode yg l meygkut solus feome fss dlm formuls persm ler. A. SASARAN UMUM Ssr umum dr perkulh dlh memberk pemhm kepd mhssw mege proses peyeles ksus fsk dlm formuls persm ler secr komputs umerk, d memberk kelelus wws tetg beberp metode dr sek byk metode yg bs dmplemetsk. B. SASARAN KHUSUS Setelh perkulh seles dlksk, mhssw dhrpk mmpu:. Memformulsk feome fss betuk persm ler ke dlm formul tertf komputs umerk.. Meyebutk beberp metode komputs umerk dlm ksus peyeles persm ler. Mejelsk perlku metode elms Guss, metode Guss-Sedel d mtrks Trdgol d dlm meg ksus persm ler yg dtg.. Megembgk pemhm deg megguk krkterstk metode-metode komputs umerk yg l. fsk-komputs 7

2 5. Meg-mplemetsk metode komputs umerk bercrk mtrks utuk persm ler dlm progrm komputer. C. URAIAN MATERI Tju sstem ler Ab, yg mempuy stu d hy stu peyeles utuk setp ss k b, d bts perht pd sstem yg mempuy jumlh persm tept sm deg jumlh vrbely, yk utuk mtrks yg koefsey A d dpt dvers-k. Sutu uj cob yg sergkl dkutp utuk meelt dpt tdky sutu mtrks dversk, ddsrk pd kosep determ. Teorem petg yg bersgkut meytk bhw mtrks A dpt dversk, jk hy jk det(a sebgm Dll Crmer yg meytk peyeles dr Ab dlm determ. Nsmu demk, determ tdk petg utuk prktek peyeles sstem ler, kre perhtug determ bsy mempuy kesult yg sm deg peyeles sstem ler. Kre ls tersebut tdk dguk determ dlm peyeles sstem ler d jug tdk perlu medefsk determ tu sedr. Metode komputs umerk utuk peyeles sstem persm ler dpt dbg dlm du jes, lgsug (drect d ters(tertve. Metode lgsug dlh metode deg tdk dy keslh pembult tu lly, k memberk peyeles yg tept dlm jumlh opers rtmetk elemeter yg terbts byky. Metode dsr yg dguk dlh elms Guss d d berbg plh metode yg bervrs dlm efses d kecermt htug. Metode ters dlh dmul deg pedekt permul megguk lgortm yg sesu, utuk medptk hsl pedekt yg lebh bk. Metode ters bervrs dlm lgortm d kecept koverges. Kelebh metode ters dlh kesederh d kesergmy dr opers yg dlkuk. Mtrks yg berkt deg sstem ler jug dgologk dlm pdt (dese tu loggr (sprse. Mtrks pdt mempuy sedkt sekl usur-usur ol, d orde mtrks tu cederug mejd reltf kecl mugk berorde tu lebh kecl. Bsy lebh efse utuk meg mslh yg melbtk fsk-komputs 8

3 mtrks semcm tu deg metode lgsug. Mtrks loggr mempuy sedkt sekl usur-usur tk ol. Bsy tmbul dr ush -ush utuk meyelesk persm dferesl deg metode selsh terhgg. Tgkt mtrks semcm mugk besr sekl, d secr del sgt cocok utuk peyeles deg metode ters. Berkut dlh beberp metode d dlm meyelesk persm ler deg pedekt mtrks, tr l:. Kdh Crmer b. Elms Guss (deg pvotg (Stblty:,Precso:Affected by Roud-off error, Bredth of Applcto:Geerl, Progrmmg Effort:Modert c. Guss Jord d. Dekomposs LU (Mtrks Spesl Trdgol (Stblty:,Precso:Affected by Roud-off error, Bredth of Applcto:Geerl, Progrmmg Effort:Mode rt e. Guss Sedel (Stblty:my ot coverge f ot dgolly domt, Precso:Ecellet, Bredth of Applcto:Approprte oly for dgolly domt system, Progrmmg Effort:Esy. Elms Guss (deg pvotg Mtrks mejd skem yg efse ketk semu koefse sstem ler Ab berd dlm deret berorde N(N. Koefse-koefse b dsmp dlm kolom N dr deret ( ytu,nb. Tp brs memut semu koefse yg dperluk utuk meytk stu persm dlm sstem ler. Mtrks legkp dytk oleh [A,b] d sstem ler tu dytk sebg berkut: [ A, b].. N.. N..... b N N.. NN b b.. b N (. fsk-komputs 9

4 Sstem Ab, dpt dselesk deg melkuk OBE (opers-opers brs elemeter pd mtrks legkp [A,b]. Vr bel-vrbel k dlh pemegg poss utuk koefse-koefse d dpt dhlgk smp khr perhtug. Opers berkut merupk opers brs elemetr yg dpt dterpk pd mtrks legkp d meghslk sstem yg setr, melput: ( Pertukr : urut du brs dpt dtukr (b Peskl : Perkl sebuh brs deg tetp tdk ol (c Peggt : Sebuh brs dpt dgtk oleh jumlh brs tu deg kelpt sebrg brs ly. Tumpu (pvotg dlh slh stu betuk peyeles elms Guss deg meetuk blg kk pd poss (k,k utuk megelms k dlm brs k,k,,n. Jk kk, mk brs k tdk dpt dpk utuk meghlgk eleme-eleme pd kolom k, d brs k hrus dtukr deg brs ly d bwh dgol utuk memperoleh eleme tumpu yg tdk ol. Jk tdk dpt dlkuk mk sstem persm tdk mempuy seles tuggl. Metode elms Guss memerluk du thp d dlm meyelesk su tu sstem persm ler. Pertm, thp elms mju (forwrd elmto bertuju megubh mtrks koefse mejd mtrks segtg ts. Kedu, dlh subttus blk (bck subttuto. Cotoh. Sstem persm umum deg, dtulsk sebg berkut b b b P( P( (. P( selesk persm ler slmult dts megguk metode elms Guss Solus Thp Pertm: Elms Mju lgkh pertm, dlh elms dr P( d P( deg sums. Defsk P d P fsk-komputs 5

5 lkuk opers-opers berkut P( P * P( d P( P * P(, mk persm ler pd (. mejd: ' ' ' ' b b b koefse-koefse j ddefsk oleh j ' b ' b j P P b j, j, ' ', P( P( P( Lgkh kedu dlh elms dr P(. Asumsk bhw Defsk P ' ' (. lkuk opers-opers berkut P( P* P( mk persm ler pd (. mejd: ' ' " b b b ' " P( P( P( koefse-koefse yg bru ddefsk oleh " b " b ' P ' P Thp Kedu: Subttus Blk b ', j,, Deg subttus blk, secr berutu ddptk, d : b ( b ( b "/ ' " ' / ' / (. (.5 Cotoh. Guk elms Guss utuk meyelesk,,, 7,,, 7,85 P( 9, P( 7, P( bw 6 gk sgfk selm komputs (.6 fsk-komputs 5

6 solus Thp Pertm: Elms Mju Opers-opers elms dlh P(,/*P( d P(,/*P( k memberk perubh pd persm.6 mejd: 7,,9,,9,, 7,85 P( 9,567 P( (.7 7,65 P( Utuk melegkp elms mju, hrus dhlgk dr P( deg opers P(,9/7,*P(, sehgg sstem tereduks mejd betuk segtg ts sebg berkut: 7,,,9,, 7,85 P( 9,567 P( (.8 7,8 P( Thp Kedu: Subttus Blk 7.8 7,, 9,567,9 (7,,5 7, 7,85,(,5,(7,, Lgkh-lgkh utuk pd cotoh. d. secr mudh dpt dmplemetsk utuk sstem persm ler yg tdk sgulr, dm mtrks segtg ts kre proses elms dtulsk, ' '. ". d persm subttus blk,. ( ' " b b ( b ' b ".. (.9 fsk-komputs 5

7 ( b ( (. Hsly kemud dsubttus blk pd persm yg ke (. Prosedurey k berulg utuk megevlus l-l, deg formul: ( ( b j j ( j j utuk,,., (. Algortm Elms Guss Pseudocode utuk mplemets elms Guss d proses subttus blk dsjk dbwh : DO k, DO k, fctor k / k,k DO jk to,j,j fctor. k,j bb fctor.bk b /, DO,, sum DO j, sumsum,j.j (b sum/,j Cotoh. Butlh progrm utuk meyelesk set persm smult dlm betuk mtrks berkut deg elms Guss! Solus fsk-komputs 5

8 /* Elms Guss */ #clude <sto.h> #clude <stdlb.h> #clude <mth.h> #defe TRUE /* [][j] : eleme mtrks, [I,j] : orde mtrks */ m( { t, j, _, _r; sttc ; sttc flot _t[][] { {,,, }, {,,, }, {,,, } }; double [][]; vod guss(; sttc t _ ; prtf ( Elms Guss \\ ; prtf ( Eleme Mtrks\ ; for ( ; <; { for ( j; j< ; j { [][j]_t[ ][j ]; prtf( %.5, [][j] ; } prtf ( \ ; } guss (, ; prtf ( Solus\ ; prtf ( \ ; prtf ( (\ ; prtf ( \ ; for ( ; <; prtf ( %5d %6.6e\,, [] [] ; prtf ( \\ ; et(; } vod guss (, t ; double [ ] []; { t, j, jc, k, kc, v, pv; r, temp, tm, v; for ( ; < (-; { for (jr ; jr<; jr{ /* elms dbwh dgol */ f ( [jr][]! { r [jr][] / [][]; for (kc ; kc< (; kc{ temp [jr][kc]; [jr][kc] [jr][kc] r* [][kc]; } fsk-komputs 5

9 } } } } for ( ; <; { /* subttus blk */ [][] [] []/[][]; for ( v ; v >; v { v [v][]; for ( kv; k < ; k { vv [v][k]*[k][]; } [v][] v/[v][v]; } retur; } Hsl progrm Eleme mtrks.e.e.e.e.e.e.e.e.e.e.e.e Solus I [] e.75e.5e Metode Guss-Sedel Dlm sub-bhs k dbhs metode peyeles sstem persm ler secr tk lgsug tu tertf. Metode perhtug secr lgsug sudh dbhs dlm sub-bhs d dep, ytu elms Guss. Metode Guss-Sedel dlh metode tertf yg secr lus telh dguk sebg ltertf metode elms. Tju stu set dr persm: [A]{X}{B}, deg sums merupk persm. Jk eleme dgol tdk ol d ly tdk dkethu, persm pertm bs dselesk sebg, persm kedu sebg d persm ketg sebg, dtujukk berkut. fsk-komputs 55

10 b (. b (.b b (.c Thp seljuty dmul proses peyeles deg memlh l cob utuk. Lgkh sederh utuk meetuk l cob deg megsumsk bhw semu l wl dlh ol. Jk dsubttusk pd persm (., mk ddptk l bru utuk b /. Kemud kt subttusk l bru d l wl berl ol utuk pd persm (.b utuk meghtug l bru. Proses dulg pd persm (.c utuk medptk l bru. Kemud kembl dulg utuk persm d prosedur berulg smp peyeles koverge cukup rpt utuk l keber. Koverges bs dcek megguk krter j j j e, % < e utuk semu, dm j d j dlh ters st tu d sebelumy. s Cotoh. Pdg sutu sstem persm,,, 7,,, 7,85 9, (.5 7, solus cu yg ber dlh,,5 d 7 Solus pertm, selesk setp persm utuk dgol yg belum dkethu 7,85,, (. 9,,, (.b 7 fsk-komputs 56

11 7,,, (.c deg sums d dlh ol, mk (. mejd 7,85,66667 l d sums, dsubttusk pd (.b memberk htug 9,,(, ,795 ters pertm dlegkp deg subttus hsl perhtug l d ke dlm persm (.c berkut 7,,(,6667,(,795 7,56 Utuk ters kedu, proses yg sm berulg d memberk hsl berkut: 7,85,(,795,(7,56,99557 e t, % 9,,(,99557,(7,56,9965 e,5 % t 7 7,,(,99557,(,9965 7,9 e t, % metode, lebh juh, koverge pd l ber. Keljut ters k memberk jwb yg lebh tept. Algortm Guss-Sedel Pseudocode utuk mplemets metode Guss-Sedel dsjk dbwh : SUBROUTINE Gsed(,b,,,m,es,lmbd DO, dummy, DO j,,j,j /dummy b b /dummy DO, sum b DO j, fsk-komputs 57

12 IF j THEN sumsum,j. j sum j ter DO setel DO, old sumb DO j, IF j THEN sumsum,j.j lmbd*sum(, lmbd*ld IF setel AND. THEN eabs(( old/ *. IF e>es THEN setel END IF terter IF setel OR (ter m EXIT END Gsed. Mtrks spesl Trdgol & Nl Ege Byk mslh terp melbtk mtrks deg kebyk elemey ol. Slh stu betuk mtrks yg eleme oly berpol dlh mtrks pt (bded mtr. Lebr pt dlh mksmum byky eleme tkol pd brsbrs sutu mtrks pt. Mtrks pt yg terkecl dlh yg lebr pty tg tu dkel sebg mtrks trdgol, sepert dtujukk pd persm (. sebg Sstem ler trdgol NN. fsk-komputs 58

13 f e g f e g f g e N f N e N g N f N... N N r r r... rn rn (. Jk elms Guss lgsug dterpk pd sstem (. mk byk opers yg sebery tdk perlu dlkuk. Agr metode lebh efse dperluk modfks. Pvotg tdk dperluk, kre pd umumy persm (. yg djump dlm prktek bersft dom secr dgol. Setelh elms k dhslk mtrks bdgol ts. Beberp metode bs dguk utuk meyelesk sstem trdgol, dtry dlh Sect, Guss Sedel d ly tergtug dr korels perlku eleme mtrks trdgol. D fsk sergkl djump ksus peyeles l ege d fugs ege dr sutu fugs ked. Akhr bhs pd stud ksus k dsggug tetg l ege d fugs ege utuk prtkel yg berd dlm sumur potesl. Metode yg efse utuk meyelesk sstem trdgol dtry dlh lgortm Thoms (Thoms Algorthm. Sepert pd dekomposs kovesl LU, lgortm terdr dr tg lgkh ytu dekomposs, subttus mju d subttus blk. Berkut dlh lgortm Thoms: ( Dekomposs DO k, ekek/fk fkfk ek.gk (b Subttus Mju DO k, rkrk ek.rk (c Subttus Blk r/f DO k,, fsk-komputs 59

14 k(rk gk.k/fk Cotoh. Selesk sstem trdgol berkut deg lgortm Thoms Solus,,, T,8 T,8 T,8, T,8 Pertm, dekomposs dmplemetsk sebg berkut e f e f e f /,,9. (,9 (,55 /,55,65, (,65 (,95 /,95,77, (,77 (, (. kemud mtrks bertrsforms mejd,,9,55,65,95,77, d dekomposs LU memberk,9 [ A ] [ L ][ U ],65,77,,55,95, Subttus mju me mberk perhtug: r r r,8 (,9,8,8,8 (,65,8,,8 (,77,,996 fsk-komputs 6

15 ,8,8 d modfks vektor,,996 yg kemud dguk dlm kojugs deg mtrks U dlm subttus blk d memberk solus, T,996 /, 59,8 T [, ( 59,8 ]/,95,58 T [,8 (,58 ]/,55 9,778 T [,8 ( 9,778 ]/, 65,97 Jwb dr lgortm Thoms bs dcek deg megguk softwre komputs populer, dlm hl dplh MATLAB (MATr LABortory sebg fslts mpuls mtrks, d sstem trdgol pd persm (. dselesk deg sgt kurt, sepert pd gmbr Gmbr. proses MATLAB dlm meyelesk sstem trdgol ::: Stud Ksus Fsk::: fsk-komputs 6

16 Arus d Tegg dlm Rgk Resstor Utuk meetuk besr rus d tegg pd rgk kombs resstor, dguk kdh Krchoff tetg rus deg formuls: d kdh Krchoff tetg tegg dlm loop: ξ R gy gerk lstrk dr sumber tegg., dm ξ dlh Tju rgk pd gmbr.. Arus dlm rgk belum dkethu bk besr mupu rhy.buk mejd persol yg rumt kre deg sums yg sederh, rh dcr pd tp lr rus. Jk hsl dr kdh Krchoff egtf, mk sums rh tetuy dperbk. Ω 5 Ω V V 5 Ω Ω Ω 5 Ω V6 V [] Gmbr. [] Rgk resstor devlus deg persm ler smult, d [b] Asums rh rus [b] Berdsrk sums pd gmbr. [b], kdh Krchoff tetg rus pd setp ode memberk: d kdh tegg pd loop dlh: 5 65 R R (. R R 5 R R 5 R 5 Lebh ljut sejumlh permslh dselesk deg em set persm dm terdpt em besr rus yg tdk dkethu, sepert terlht pd pemodel fsk-komputs 6 (.5

17 mtrks. Dsmpg tdk prkts dselesk deg tg, sstem mt mudh dtg deg metode elms Seljuty dlm ksus, solus ddptk: 65 6,58 6,58 5 5,65,585,585,585 besr d rh rus d tegg pd ode d loop dtujukk pd gmbr dbwh: 5,85 V,585 69, V 6,58 V V 6,5 V V6 V,8 V 6 Gmbr. Besr d rh rus d tegg pd rgk resstor Lebh ljut, deg megguk lgortm komputs umerk d pemrogrm komputer, tpe ksus sepert mejd lebh sederh. Nl Ege d Fugs Ege pd Sumur Potesl Solus Persm Schrodger Dmes Stu dlm sstem Kutum fsk-komputs 6

18 V V Persm Schrodger tk tergtug wktu mejd cu dlm ksus. Vo b b mk dperoleh ψ( m ( E V ψ( h (.6 dm ψ ( dlh fugs ege, d E dlh l ege. Dlkuk ormlss de g mesubttusk: h h E, V v d by, mb mb ψ( y ( v ψ( y (.7 y deg megubh persm (.7 ke dlm betuk komputs umerk, memberk persm ters berkut: ψo h ( v ψ ψ (.8 ψ h v ψ ψ (.9 ( d seterusy hgg ψ h ψ ψ (. ( v dm v dlh potesl d ttk. Deg meerpk syrt bts ψ pd b tu y d o ψ pd b tu y, mk ddptk betuk l dr persm (.8, (.9 d (. ytu sstem trdgol sebg berkut, fsk-komputs 6

19 fsk-komputs 65 ( ( ( ( ( v h v h v h v h v h (. deg mtrks fugs sebg berkut :. ψ ψ ψ ψ ψ (. Perkl mtrks koefse dlm sstem trdgol deg mtrks fugs sm deg ol. Peyeles mtrks dts k trvl jk determ mtrks plg kr tdk sm deg ol. Agr tdk trvl mk determ tdk boleh sm deg ol. Determ mtrks dpt dhtug deg cr membut sub-sub determ yg dhtug sebg berkut: ( v h P (.. ( P v h P (. d seterusy hgg dperoleh tur umum utuk mecr setp sub determ, ytu. ( P P v h P (.5 Sehgg determ keseluruh dr mtrks pd persm (. dlh C P P (.6 Persm (.6 merupk polom sehgg utuk meyelesky dpt dguk metode Sect dlm mecr kr-kr sebuh polom, sehgg

20 fsk-komputs 66 ddpt l-l yg memeuh persm (.6. Adpu metode Sect dlm formul tertfy dlh ( ( ( P P P (.7 Nl-l yg dperoleh dr persm (.7 merupk l-l ege dr prtkel-prtkel pd ksus sumur potesl. Utuk setp l dsubttusk ke mtrks (., d deg megguk metode Guss Sedel bs dperoleh fugs ege gelombg utuk terkt. Adpu subttus l wl utuk metode Guss Sedel (sebgm lzmy metode Guss Sedel dberk l d seterusy, hgg khry dperoleh stu buh l cob yg dpt memberk l fugs-fugs gelombg yg terorlk. Deg demk proses peyeles secr komputs umerk memeuh persyrt peyeles sebgm peyeles ltk utuk persm Schrodger. D. SOAL_SOAL (. Selesk sstem segtg ts berkut : (. Crlh prbol yabc yg mellu tg ttk: (,, (,7 d (, (. Megguk elms Guss deg pvotg selesk sstem persm ler berkut: (. Mul deg semu l wl ol guk ters Guss-Sedel utukmecr l, y d z smp ters ke-tg pd mtrks berkut.

21 5 y z 8 y z y z Apkh ters Guss Sedel k koverge ke seles? (.5 Butlh progrm utuk stud ksus pertm d kedu, deg metode yg dmksud. E. DAFTAR PUSTAKA Chpr, S.C., d Cle, R.P., Numercl Methods for Egeers, McGrw-Hll, 998 Jmes, M.L., G.M. Smth, d J.C. Wolford, Appled Numercl Methods for Dgtl Computtos, rd ed. Hrper & Row, 985 Koo, S.E., Computtol Physcs, Addso-Wesley Ic, 986 Mthews, J.H., Numercl Methods for Mthemtcs, Scece d Egeerg, Pretce-Hll Ic., 99 McCrcke, D. D., Computg for Egeers d Scetsts wth Fortr 77, Wley, 98 Morrs,J.L., Computtol Methods Elemetry Numercl Alyss, Wley, 98 Nkmur, S., Appled Numercl Methods C, Pretce-Hll Ic. 99 Ykowtz, S., d F. Szdrovszky, A Itroducto to Numercl Computtos, Mcmll, 986 fsk-komputs 67