Tgs besr Metode merk Mege : cotoh sol-sol metode merk d pembhsy Nm ggot : Abdl hrrs hdyt (95 Are krw (95 Yog tr wrme (959 Dose : Her dbyolksoo.mt Jrs tekk elektro Fklts tekk Uversts dls Pdg
Bb Dsr teor A. Akr-kr persm. Metode terttp.. metode grk Utk memperoleh tksr kr persm ( = lh deg membt grk gs t d megmt dm memotog smb. Ttk, yg meytk hrg tk ( =, memberk st pedekt ksr dr kr tersebt. Cotoh.. Pedekt grk. Gkpedektgrk tkmemperolehstkr persmdr ( =e -. Solsy dlhsebg berkt: X,,,,6 F(,,69,7 -,5 Gmbr. Gmbr.. Ilstrs pedekt grk tk memechk persm ljbr d trsedetl. Grk ( = e - terhdp. Akr ses deg hrg dm ( =, yt ttk dm gs memotog smb. Pemerks secr vsl mege plot
memberk tksr ksr,57. Hrg sebery dlh,5679 Tekk grk prkts dgk, d dpt memberk tksr kr secr ksr, tp tdk press. Dgk sebg tebk wl dlm metode merk. Iterprets grk petg tk memhm st-st gs d dpt memperkrk jebk pd metode merk, sepert terlht pd gmbr. d bwh. Gmbr. memperlhtk sejmlh cr dm kr bs berd dlm tervl yg djelsk oleh st bts bwh d bts ts b. Gmbr.b memperlhtk kss dm sebh kr tggl dkrg oleh hrghrg post d egt dr (. Gmbr. Gmbr.. Ilstrs sejmlh cr yg mm bhw sebh kr bs terjd dlm sebh tervl yg djelsk oleh bts bwh d bts ts b. Bg ( d (c mejkk bhw bl ( d (b mempy td yg sm, tdk k d kr-kr t kr dlm jmlh gep pd tervl. Bg (b d (d mejkk bhw bl gs mempy td yg berbed pd ked ttk jg, k terdpt kr dlm jmlh gjl pd tervl. Tetp gmbr.d, dm ( d (b berlw td terhdp smb, memperlhtk kr yg berd d dlm tervl. Ummy jk ( d (b mempy td yg berbed k terdpt kr yg jmlhy gjl dlm tervl. Sepert dtjkk oleh gmbr. d c, jk ( d (b mempy td yg sm, tdk terdpt kr-kr t kr yg jmlhy gep berd dtr hrg-hrg t. Meskp geerlss bsy ber, m terdpt kss-kss dm hl t tk dpt dpegg. msly kr gd. Yk gs yg meyggg smb (gmbr. d gs- gs dskot (gmbr.b bs meylh prsp. Gmbr.. Ilstrs beberp perkecl terhdp kss-kss mm yg dtjkk dlm gmbr.. ( kr gd yg terjd sewkt gs meyggg smb. Dlm hl, wlp ttk-ttk jgy berlw td, terdpt kr-kr
dlm jmlh gep tk tervl tersebt. (b gs dskot dm ttk-ttk jg td yg berlw jg megrg kr-kr dlm jmlh gep. Strteg khss dbthk tk peet kr-kr dlm kss. Sebg cotoh gs yg mempy kr gd dlh persm kbk ( = ( ( (. Perhtk bhw = membt ked sk poloml t sm deg. Jd = dsebt sebh kr gd...metode bg d sepert yg telh dsggg sebelmy, metode bseks merpk slh st cotoh dr metode kolde. Metode bseks jg dsebt metode pembg tervl kre membg re tr blg yg merpk tebk wl mejd bg, dr d bg dplh bg m yg megdg kr d bg yg tdk. Bg yg tdk megdg kr k dbg seljty pecr dlkk pd bg yg dperkrk megdg kr.hl dlkk berlg-lg hgg dperoleh kr persm. Lgkh-lgkh meetk kr deg metode bseks. Meetk d ttk l wl, msly sebg bts bwh d b sebg bts ts. Nl dperoleh deg cr meebk. Nm hrs dshk rge tebk sedekt mgk deg perkr kr persm.. Dr l wl tersebt, tetk l tegh ( c deg rms : c Sehgg k mcl d derh yt derh kr ( yt tr c deg ( <<c d tr c deg b (c<<b. Setelh memperoleh l c, mskk l tersebt pd persm ( sehgg persm mejd (c. Jk (c > mk kr( berd pd derh < < c t d sebelh kr grs bg derh (grs yg dtrk vertcl dr ttk c
Jk (c < mk kr ( berd pd derh c < < b t d sebelh k grs bg derh (grs yg dtrk vertkl dr ttk c Derh kr bl (c > c b Derh kr bl (c < 5. Jk (c > mk = tetp mejd bts bwh semetr c = bts ts yg br. Ulg lgkh hgg medptk l tegh tr d c, seljty kt sebt l tegh deg sj. Jk (c < mk c = bts bwh semetr b = tetp bts ts. Ulg lgkh hgg medptk l tegh tr c d b, seljty kt sebt l tegh deg sj. 6. Setelh medptk l tegh ( lg lgkh d smp ( = t medekt (tergtg tolers yg dberk. Bl ( = t medekt berrt kr persmy sdh dtemk sehgg ters dhetk. Kelebh d kekrg metode bseks Kelebh : metode bseks lebh mdh dlm pembt progrmy d sell mecp lj yg koverge ke rh kr yg dgk.
Kekrg : l wl yg hrs dtebk lebh byk d pecr kr pedekt membthk wkt lebh lm kre begt kovergey sehgg memerlk byk ters. Ctt bl tolers error yg kt tetk pd sebh metode bseks semk kecl, msly. (tp lebh kecl lg mk byky ters yg dbthk k semk bertmbh besr. Hl tersebt kre semk kecl l tolers, mk k semk telt hsl yg dgk. Metode terbk..iters st ttk sederh Metode st ttk sederh (ed pot Metode ters sederh dlh metode yg memshk dr ( sehgg dperoleh : = g(. Kemd dtetk hrg o d ml meghtg = g( o, = g( d setersy. Secr mm ters dpt dtlsk sebg berkt: X + = g( dm =,,,... Peyeles persm d ts dsebt metode st ttk sederh. Hsl dlh peyeles dr ( kre betk = g( dpt dkemblk ke betk sly yt (. Kt bs sj memperoleh beberp betk g(. Hl sgt berkt deg proses pemsh secr ljbr. Cotoh: Dketh persm ( = - + =. Deg meggk rms bc dpt dketh bhw l, =.5 ±.5 sehgg =.68 d =.8966. Deg megeth l eksk dr kr-kr persm ( kt dpt meghtg keslh tersy. Peyeles deg metode st ttk sederh; ( dbh mejd: = g( = ⅓ ( + sehgg + = ⅓ ( + Jk kt memlh =, kt k medpt rt perhtg: Iters X X X. X.667
X.8 X. X.9 Hsl mejkk bhw ters k mej ke l yg kecl. Jk kt memlh = mk kt k medptk kods yg sm. Tetp jk kt memlh = mk hsly dlh: Iters X X X. X. X.7 X 5.766 X.5 Nl k semk membesr deg bertmbhy ters (dverge. Utk medptk hsl yg koverge, mk persm = g( hrs dmodks. = g ( = (/ sehgg + = (/. D jk kt memlh = mk kt k medptk rt l sebg berkt: Iters X X X. X. X.5 X.6 X.65 Dm hsl mejkk kecederg tk mej kl tg besr. Jk kt megbh l wl ( mk hsl yg ddptk dlh : Iters X X X. X.667 X.65 X.69 X.68
Hsl lebh mempy keslh ters yg lebh kecl dbdgk deg sms sebelmy. Grk d bwh mejkk proses ters yg telh dbhs d ts. gmbr.6 proses ters metode st ttk sederh. metod ewto rphso Metode ewto t yg bs dkel deg metode ewto rphso dpt dgk tk mecr kr dr st gs. Keggl metode dlh memlk lj koverges kdrtk, sehgg metode lebh cept tk koverge mej kr pedekt drpd metode l yg memlk lj koverges ler. Pecr kr dlht dr t grde grk st gs persm (tr gs persm. Pd dsry, lgortm metode ewto tk mecr kr st gs ( dml deg meetk l wl ters terlebh dhl, mslk =. Pd setp ters, metode ewto k mecr st l ktklh b yg berd pd smb-. Nl b dperoleh deg merk grs sggg gs ( d ttk = ke smb-. Metode ewto rphso dlh metode pedekt yg meggk st Ttk wl d medekty deg memperhtk slope t grde pd ttk tersebt.ttk pedekt ke + dtlsk deg : + = - ( ( metode ewto rphso dpt dgmbrk sebg berkt :
Lgkh-lgkh meetk kr deg metode ewto-rphso deg meggk sebg tebk wl, dljtk deg mecr ttk (, (. Kemd dbt grs sggg dr ttk (, (, sehgg dperoleh ttk potog (, tr smb- d grs sggg ttk (, (. Kemd dljtk lg deg mecr ttk (, (. Dr ttk (, ( kemd dbt grs sggg, sehgg dperoleh ttk potog (, tr smb- d grs sggg ttk (, (. Demk setersy. Utk lebh jelsy, perhtk lgkh berkt. t (bet = ( =, Mk ters pertm X = - Dljtk ters ked = - ( '( D setersy, deg cr yg sm ddpt + = - '( ( ters dhetk jk d ters yg berrt meghslk hmpr kr yg sm (jk selsh tr kr-kry relt sm t + = < tol ( '( ( (
Dr betk (, terdpt peyebt (. Sehgg gr setp ters tdk terjd keslh (error, mk selm ters l ( tdk boleh ol, kre pembg tdk boleh ol. Kelebh d kekrg metode ewto-rphso Kelebh : bl tksr wl kebetl memg medekt kr yg sesgghy mk wkt yg dbthk tk meghtg kr lebh cept. Kekrg : bl tksr wl tdk tept, hsly jstr k dverge (semk mejh l kr yg sebery. Permslh pd pemk metode ewto-rphso. Metode tdk dpt dgk ketk ttk pedekty berd pd ttk Ekstrm t ttk pck, kre pd ttk l ( = sehgg l peyebt dr ( F ( sm deg ol, secr grs dpt dlht sebg Berkt:
pedekt pd ttk pck bl ttk pedekt berd pd ttk pck, mk ttk seljty k berd d tk berhgg.. Metode mejd slt t lm medptk peyeles ketk ttk pedekty berd d tr d ttk stsoer. ttk pedekt berd dtr ttk pck bl ttk pedekt berd pd d ttk pck k dpt megkbtk hlgy peyeles (dverges. Hl dsebbk ttk seljty berd pd slh st ttk pck t rh pedekty berbed. Peyeles permslh yg tmbl pd metode ewto-rphso
Utk dpt meyelesk ked permslh pd metode ewto rphso, mk metode ewto rphso perl dmodks deg :. Bl ttk pedekt berd pd ttk pck mk ttk pedekt tersebt hrs d geser sedkt, = ±δ dmδ dlh kostt yg dtetk deg demk ( d metode ewto rphso tetp dpt berjl.. Utk meghdr ttk-ttk pedekt yg berd jh, sebky pemk metode ewto rphso ddhl oleh metode tbel, sehgg dpt d jm koverges dr metode ewto rphso.. Metode sect. mslh yg ddpt dlm metode ewto-rphso dlh terkdg slt medptk tr pertm, yk (. Sehgg deg jl pedekt ' mejd y ( ( ( ( Persm d ts memg memerlk tksr wl, tetp kre ( tdk membthk perbh td dtr tksr mk sect bk metode lokde. Gmbr 5. Tekk serp deg tekk ewto-rphso dlm rt bhw st tksr kr drmlk oleh ekstrpols sebh grs sggg dr gs terhdp smb. Tetp metode sect lebh meggk deres drpd tr tk memperkrk kemrg/slope Perbed metode sect d regl ls.
Persm d metode sect mp regl ls detk sk dem sk. Kedy meggk tksr wl tk meghtg proksms slope gs yg dgk tk berproyek terhdp smb tk tksr br kr. Perbedy pd hrg wl yg dgtk oleh tksr br. Dlm regl ls, tksr terkhr kr meggtk hrg sl m sj yg megdg st hrg gs deg td yg sm sepert (r. Sehgg tksr sets megrg kr. Sect meggt hrg-hrg dlm deret yg kett, deg hrg br + meggtk, d meggtk -. Sehgg hrg terkdg dpt terletk pd rs kr yg sm. Pd kss tertet bs dverge. Pd gmbr grk d bwh dsjk pegg metode regl ls d sect tk Meksr kr ( = l, dml dr hrg = - =,5 d X = = 5,: Gmbr 5.. Perbdg metode regl ls d sect. Iters pertm ( d (b tk ters ked metode dlh detk. Tetp pd ters ked (c d (d, ttk yg dpk berbed.
Gmbr 5.. B. Sstem persm ljbr ler. Elems gss Metode elms gss merpk metode yg dkembgk dr metode elms, yt meghlgk t megrg jmlh vrbel sehgg dpt dperoleh l dr st vrbel bebs. Cr elms sdh byk dkel. Elms gss dlh st cr megopersk l-l d dlm mtrks sehgg mejd mtrks yg lebh sederh. Cry dlh deg melkk opers brs sehgg mtrks tersebt mejd mtrks yg eselo-brs. dpt dgk sebg slh st metode peyeles persm ler deg meggk mtrks.cry deg megbh persm ler tersebt ke dlm mtrks tergmets d megopersky.setelh mejd mtrks eselo-brs, lkk sbstts blk tk medptk l dr vrbel-vrbel tersebt. Metode bergkt dr keyt bhw bl mtrks berbetk segtg ts (meggk opers brs elemeter sepert system persm berkt : [ ] [ ] [ ] Mk solsy dpt dhtg deg tekk peylg mdr sbsttto: (bckwrd
( Sekl dketh, mk l dpt dhtg deg: Kods sgt petg. Sebb bl, persm dts mejerjk pembg deg ol. Apbl kods tersebt tdk dpeh, mk spl tdk mempy jwb.. Metode gs jord Dlm ljbr ler, elms gss-jord dlh vers dr elms gss.pd metode elms gss-jord kt membt ol eleme-eleme d bwh mp d ts dgol tm st mtrks.hsly dlh mtrks teredks yg berp mtrks dgol st (sem eleme pd dgol tm berl, eleme-eleme ly ol. Dlm betk mtrks, elms gss-jord dtls sebg berkt. [ ] [ ] Solsy: Sepert pd metode elms gss ï, metode elms gss-jord ï tdk meerpk tt-cg pvotg dlm proses elmsy. Lgkh-lgkh opers brs yg dkemkk oleh gss d dsemprk oleh jord sehgg dkel deg elms gss-jord, sebg berkt:
. Jk st brs tdk selrhy dr ol, mk blg tk ol pertm pd brs t dlh. Blg dsebt tm (ledg.. Jk terdpt brs yg selrhy terdr dr ol, mk brs-brs k dkelompokk bersm pd bg plg bwh dr mtrks.. Jk terdpt d brs berrt yg tdk selrhy dr ol, mk tm pd brs yg lebh redh terdpt pd kolom yg lebh k dr tm pd brs yg lebh tgg.. Setp kolom memlk tm memlk ol pd tempt l. Algortm metode elms gss-jord dlh sebg berkt:. Mskk mtrks d vector b besert kry. Bt gmeted mtrks [b] mk deg. Utk brs ke- dm = s/d Perhtk pkh l sm deg ol: Bl y: Pertkrk brs ke- d brs ke +k, dm tdk sm deg ol, bl tdk d berrt perhtg tdk bs dljtk d proses dhetk deg tp peyeles. Bl tdk: ljtk b Jdk l dgoly mejd st, deg cr tk setp kolom k dm k= s/d +, htg. Utk brs ke j, dm j=+ s/d Lkk opers brs elemeter tk kolom k dm k= s/d Htg Htg 5. Peyeles, tk = s/d (bergerk dr brs ke smp brs pertm. Metode gss sedl Metode ters gss-sedel Metode ters gss-sedel dlh metode yg meggk proses ters Hgg dperoleh l-l yg berbh. Bl dketh persm ler smlt:
Berk l wl dr setp (= s/d kemd persm ler smlt dts Dtlsk mejd: Deg meghtg l-l (= s/d meggk persm-persm d ts Secr ters-meers hgg l tk setp (= s/d sdh sm deg l Pd ters sebelmy mk dperoleh peyeles dr persm ler smlt Tersebt. At deg kt l proses ters dhetk bl selsh l (= s/d Deg l pd ters sebelmy krg dr l tolers error yg dtetk. Ctt: Ht-ht dlm meys sstem persm ler ketk meggk metode ters Gss-sedel. Perhtk setp koese dr msg-msg pd sem Persm d dgol tm (. Letkk l-l terbesr dr koese tk Setp pd dgol tm. Mslh dlh mslh pvotg yg hrs ber Ber dperhtk, kre peys yg slh k meyebbk ters mejd Dverge d tdk dperoleh hsl yg ber.. Dekomposs l..metode crot Dekomposs l metode crot
Utk spl deg =, persm.5. dpt dtlsk sebg: l l l l l l l l l l l l l l l l.9 Metode crot dtrk deg meggk perkl mtrk. Utk meghtg rs kr persm.9 ll meymky deg rs k. Lgkh pertm dlh meglk brs pertm [l] deg kolom pertm []. Lgkh meghslk: L = l = l = l = Secr mm dpt dtlsk bhw: L l = tk =,,.,.. Seljty brs pertm [l] dklk deg kolom kolom dr [] tk memberk: L = l. = l. U = l. U = Hbg pertm sdh jels, d berkty dlh: l l At secr mm dytk: j j tk j,,, l l..b Seljty brs ked smp keempt dr [l] dklk deg kolom ked [] sehgg meghslk: L. + l = l. + l = l. + l = Msg msg dpt dpechk tk,, d l : L = l l. tk =,,.., Klk brs ked [l] deg kolom kolom ketg d keempt:..c L. + l. = l. + l. = Yg dpt dpechk tk d : l. At secr mm: l l l.
j l. j j tk j,,, l..d Dr hsl-hsl d ls mk dpt dberk rms mm metode dekomposs cr crot: Cotoh: Lkk dekomposs l dr spl deg metode crot:. 5. + = +. = 8.. +. = 6 Deg memk rms yg d: L = l =- l Brs pertm dr []: l l 5,5,5 Kolom ked [l]: l l l l.. Eleme terkhr dr []: l. ( (,5,5 ((,5,5 ( (,5,5 l Eleme terkhr dr [l]: L = l. l. = (,5,5 (- = Jd dekomposs l dlh: L,5,5 U,5,5 D bl hsl dekomposs dgk tk meyelesk spl tersebt, mk lgkh seljty dlh sebg berkt (ses gmbr.
6 8,5,5 d d d C D L Deg melkk sbstts mj:,5( (6 6..,5 6 ( 8. 6 l d l d l c d l d l c d d l l Kemd []{}={d} 6,5,5 Deg sbstts mdr k dperoleh:,5(,5( ( 6. (.. d d d..metode dooltle metode doolttle merpk sebh lgortm ktorss l yg mesyrtk eleme- eleme pd dgol tm mtrk l berl, sehgg mtrk l berbetk: l = Algortm peyeles sstsebgberkt:m persm ler meggk metode doolttle dlh sebg berkt :. Fktorss,tetpkmtrkld,sehgg=l.dsldlhmtrksegtgbwh (mtrksegtgbwhdegelemedgol,sedgkdlhmtrksegtgt s.. Meghtg eleme- eleme pd mtrk deg rms : L L L L L L
Utk j = r,r +,..., Dkt meghtg eleme- eleme mtrk l deg rms : Utk = r +,r +,...,. Desky=.,tetpkhrgydrpersmlerl. y=b.dptdlkk degljbrbstphrsmelkkopersversterhdpmtrkl.. Setelhttetpkdrpersm.=y.dsjgtdkdperlkopersverster hdp mtrk. Cotoh : Tetk sols dr persm ler d bwh :
Peyeles : Megbh persm d ts mejd mtrk : [ ] [ ] Mtrk dktork mejd mtrk l d mtrk, dm mtrk l merpk mtrk segtg bwh deg dgol tmy berl [ ] [ ] [ ] = l Deg meglk ked mtrk d rs k mk dperoleh mtrk d bwh : [ ] [ ] Seljty yk meymk mtrk pd rs kr deg mtrk pd rs k: sehgg sehgg sehgg sehgg sehgg ( sehgg Hsl yg telh dperoleh d ts kemd dmskk ke persm =l dlm betk mtrk mejd : [ ] = [ ] [ ] = l Dr persm. X = b dpt djbrk mejd : L.. = b L. Y = b
Dr persm l.y = b dptkt htg y, y d y. [ ] [ ] [ ] Sehggdperoleh: l y = b [ ] [ ] Setelhkolomy dtemk,l l, d dptpldhtgdrpersm: [ ] [ ] [ ] Sehggdperoleh : = y [ ] [ ]..Metode cholesky dlm dekomposs cholesky, mtrks koese [] dpt ddekomposs mejd betk: [] = [l] [l] t. Sk-sk dlm persm. dpt dperoleh deg cr yg sm sepert cr crot. Hsly dpt dytk dlm hbg: l k k lj j l. l kj tk,,..., k -. l kk kk k j l kj. Cotoh: Gk dekomposs cholesky tk mtrks koese berkt :
A 6 5 55 5 55 5 55 5 979 Utk brs ke st, deg meggk.: l 6,95 Utk brs ked, d persm.: l l 5,95 Dr. memberk: 6,7 l l 55 (6,7,8 Utk brs ketg, persm. (k= memberk: l l l l l 55,95. l D eleme terkhr [l] dlh:,5 5 6,7(,5,96,8 l l l 979 (,5 (,96 6,6 Sehgg dekomposs l metode cholesky meghslk: A 6 5 55 5 55 5 55,95 5 6,7 979,5,8,96,95 6,6 6,7,8,5,96 6,6 D. Pecocok krv. Regres kdrt terkecl Iterpols regres Regres dlh sebh tekk tk memperoleh persm krv pedekt dr ttk-ttk dt Regres ler Regres ler dgk meetk gs ler (grs lrs yg plg ses deg kmpl ttk dt (,y yg dketh.
Dlm regres ler yg dcr dlh l m d c dr gs ler y=m+c, dm: N N N N N N y y N m m y N m N y c N N Regres ekspoesl Regres ekspoesl dgk meetk gs ekspoesl yg plg ses deg kmpl ttk dt (,y yg dketh. Regres ekspoesl merpk pegembg dr regres ler deg memtk gs logrtm. Perhtk : Y = e +b Deg melogrtmk persm d ts k dperoleh: Ly = l(e +b Ly = + b At dpt dtlsk bhw: Z = + b dm z = l y Deg demk dpt dgk regres ler dlm meetk gs ekspoesl yg plg ses deg dt.
Regres poloml Regres poloml dgk meetk gs polyoml yg plg ses deg kmpl ttk dt (,y yg dketh. Fgs pedekt : Y = + + +... + Regres poloml tgkt dkembgk dr model mtrk orml sebg berkt: y y y y. Iterpols Pegert terpols- terpols poloml Iterpols dlh st cr tk mecr l d tr beberp ttk dt yg telh dketh. Dlm kehdp sehr- hr,terpols dpt dgk tk memperkrk st gs dm gs tersebt tdk terdes deg st orml, tetp ddesk hy deg dt- dt t tble yg tersed. Ad berbg mcm terpols berdsrk gsy, d try dlh terpols ler, terpols kdrt, d terpols poloml. Deg berbg mcm metode tr l metode eto d metode lgrge, m ds kt k membhs deg metode ewto. Terdpt perbed tr terpols deg ekstrpols. Berkt pejelsy gr dpt dphm Iterpols poloml (polom
Adlh sebh metode tk meksr (megestms l d tr ttk- ttk dt yg tept.persm poloml dlh persm ljbr yg hy megdg jmlh dr vrbel berpgkt blg blt (teger. Betk mm persm poloml order dlh: ( = + + + + (. Deg,,,, dlh prmeter yg k dcr berdsrk ttk dt, dlh derjt (order dr persm poloml, d dlh vrbel bebs. Utk ( + ttk dt, hy terdpt st t krg poloml order yg mell sem ttk. Msly, hy d st grs lrs (poloml order yg meghbgk d ttk (terpols ler gmbr., demk jg tg bh ttk dpt dhbgk oleh gs prbol (terpols kdrt gmbr.b, sedg tk ttk(terpols kbk gmbr.c. Iterpols polom terdr ts peet polom k orde ke- yg cocok deg + ttk dt. Wlp terdpt st, d hy st, polom orde ke- yg cock deg + ttk, terdpt bergm betk mtemtk tk peggkp polom tersebt. Iterpols ler Sebh betk terpols yg plg sederh deg meghbgk d bh ttk dt deg grs lrs. Lht gmbr berkt Gmbr.terpols poloml
Gmbr. Iterpols ler Dketh l st gs d ttk d, yt ( d (. Deg metode terpols ler k dcr l gs d ttk, yt (. Ideks pd ( mejkk bhw terpols dlkk deg terpols poloml order st. Dr d segtg sebg bc d de sepert tmpk dlm gmbr, terdpt hbg berkt: BC AB DE AD ( ( ( ( ( ( ( ( ( (. Persm (. dlh rms terpols ler, yg merpk betk terpols poloml order st. Sk [ ( ( ] / ( dlh kemrg grs yg meghbgk d ttk dt d merpk perkr bed hgg dr tr pertm. Semk kecl tervl tr ttk dt, hsl perkr k semk bk. Cotoh sol : Dcr l l deg metode terpols ler berdsr dt l = d l 6 =,797595. Htg jg l tersebt berdsr dt l d l =,869. Utk membdgk hsl yg dperoleh, dhtg besr keslh (dketh l eksk dr l =,6978. Peyeles: Deg meggk persm (., dhtg deg terpols ler l l pd = berdsr l l d = d = 6.
( ( ( ( ( ( = +,797595 6 ( =,5859. Besr keslh dlh: e t =,6978,6978,5859 % = 8, %. Apbl dgk tervl yg lebh kecl, yt l = d =, mk:,869 ( = + ( =,698. Besr keslh dlh: e t =,6978,6978,698 % =, %. dr cotoh mpk bhw deg meggk tervl yg lebh kecl ddpt hsl yg lebh bk (keslh lebh kecl. Gmbr, mejkk ked terpols bersm deg gs yg sebery. Gmbr. Iterpols ler pelks grs mecr l Iterpols kdrt Terdpt strteg tk memperbk tksr deg memperkelk st kelegkg ke grs yg meghbgk ttk- ttk tersebt. Deg tersedy tg bh ttk dt, mk dpt dlksky deg polom orde ked (polom kdrt. ( = b + b ( + b ( ( (.
Meskp tmpky persm (. berbed deg persm (., tetp sebery ked persm dlh sm. Hl dpt dtjkk deg meglk sk-sk persm (. sehgg mejd: ( = b + b b + b + b b b At ( = + + Deg = b b + b = b b b = b Terlht bhw persm (. sm deg persm (.. Seljty tk keperl terpols, persm poloml dtls dlm betk persm (.. Berdsrk ttk dt yg d kemd dhtg koese b, b,d b. Berkt dberk prosedr tk meetk l dr koese-koese tersebt. Koese b dpt dhtg dr persm terpols kdrt, deg memsk l =. ( = b o + b ( o + b ( ( b o = ( (. Bl persm (. dsbsttsk ke dlm persm (., kemd dmskk ke dlm l =, mk k dperoleh koese b : ( = ( + b ( + b ( ( ( ( b = (.5 Bl persm dts dsbsttsk ke dlm persm (. d l =, mk k dperoleh koese b : ( ( ( = ( + ( + b ( ( b ( ( = ( ( = ( ( = ( ( ( ( ( ( ( ( [( + ( ] ( ( + ( (
At ( b = ( ( ( ( ( ( ( ( ( ( b = (.6 Deg memperhtk persm (., persm (., persm (.5 d persm (.6 terlht bhw d sk pertm dr persm (. dlh ekvle deg terpols ler dr ttk ke sepert yg dberk oleh persm (.. Sedgk sk terkhr, b ( ( merpk tmbh kre dgky krve order. Koese b d b dr terpols poloml order persm (.5 d persm (.6 dlh mrp deg betk bed hgg tk tr pertm d ked, deg demk peyeles terpols poloml dpt dlkk deg meggk betk bed hgg. C. Itegrs merk Metode tegrs smpso D smpg meggk rms trpesm deg tervl yg lebh kecl, cr l tk medptk perkr yg lebh telt dlh meggk poloml order lebh tgg tk meghbgk ttk-ttk dt. Msly, pbl terdpt st ttk tmbh d tr ( d (b, mk ketg ttk dpt dhbgk deg gs prbol (gmbr. Apbl terdpt d ttk tmbh deg jrk yg sm tr ( d (b, mk keempt ttk tersebt dpt dhbgk deg poloml order tg (gmbr b. Rms yg dhslk oleh tegrl d bwh poloml tersebt dkel deg metode (tr smpso.
Gmbr tr smpso.. Atr-tr smpso... tr smpso / Gmbr per metode smpso D dlm tr smpso / dgk poloml order d (persm prbol yg mell ttk (, ( d ( + tk medekt gs. Rms smpso dpt dtrk berdsrk deret tylor. Utk t, dpdg betk tegrl berkt. I( ( d (persm Apbl betk tersebt ddereslk terhdp, k mejd: di( I' ( ( (persm d
Deg memperhtk gmbr d persm ( mk persm deret tylor dlh: ''(! Δ '(! Δ ( Δ ( Δ ( ( I I I (Δ '''(! Δ 5 O (persm ''(! Δ '(! Δ ( Δ ( Δ ( ( I I I (Δ '''(! Δ 5 O (persm Pd gmbr, l ( + dlh ls dbwh gs ( tr bts d +. Sedgk l ( dlh ls tr bts d (. Deg demk ls d bwh gs tr bts d + yt (, dlh ls ( + dkrg ( t persm ( dkrg persm (. A = ( + ( At (Δ ''( Δ ( Δ 5 O A (persm 5 Nl ''( dtls dlm betk deresl terpst: ( Δ Δ ( ( ( ''( O Kemd betk dts dsbsttsk ke dlm persm 5. Utk memdhk pels, seljty ots ( dtls dlm betk, sehgg persm 5 mejd: (Δ (Δ Δ ( Δ Δ 5 O O A At (Δ ( Δ 5 O A (persm 6
Persm 6 dkel deg metode smpso /. Dber tmbh m / kre dbg deg. Pd pemk st ps, dlm betk: b, sehgg persm 6 dpt dtls b A ( ( c ( b (persm 7 6 Deg ttk c dlh ttk tegh tr d b. Keslh pemotog yg terjd dr metode smpso / tk st ps dlh: 5 t Δ 9 ''''( Oleh kre b, mk: ( b t 88 Cotoh sol: 5 ''''( Htg I e d, deg tr smpso /. Peyeles: Deg meggk persm 7 mk ls bdg dlh: b A 6 6 ( ( c ( b ( e e e 56,7696. Keslh terhdp l eksk: 5,5985 56,7696 t 5,5985 % 5,97 %.... Atr smpso / deg byk ps Sepert dlm metode trpesm, metode smpso dpt dperbk deg membg ls dlm sejmlh ps deg pjg tervl yg sm (gmbr 5: b
deg dlh jmlh ps. gmbr 5 metode smpso deg byk ps Ls totl dperoleh deg mejmlhk sem ps, sepert pd gmbr 5. b ( d A A... A (persm 8 Dlm metode smpso jmlh tervl dlh gep. Apbl persm 6 dsbsttsk ke dlm persm 8 k dperoleh: b Δ Δ Δ ( d ( (... ( At b Δ ( d ( ( b ( ( (persm 9 Sepert pd gmbr 5, dlm pegg metode smpso deg byk ps jmlh tervl dlh gep. Perkr keslh yg terjd pd tr smpso tk byk ps dlh: ( b 8 5 '''' Deg '''' dlh rert dr tr keempt tk setp tervl. Cotoh sol: Htg I e d, deg metode smpso deg =. Peyeles: Deg meggk persm 9 mk ls bdg dlh:
I [ e e ( e e e Keslh terhdp l eksk: 5,5985 5,8686 t 5,5985 ] % 5,8686.,5%.... Metode smpso /8 Metode smpso /8 dtrk deg meggk persm poloml order tg yg mell empt ttk. I b b ( d ( d Deg cr yg sm pd per tr smpso /, khry dperoleh: Δ I 8 ( ( ( ( (persm Deg: b Persm dsebt deg metode smpso /8 kre dklk deg /8. Metode smpso /8 dpt jg dtls dlm betk: ( ( ( ( I ( b (persm 8 Metode smpso /8 mempy keslh pemotog sebesr: t Δ 8 ''''( (persm b Meggt, mk: ( b t 68 5 ''''( (persm b Metode smpso / bsy lebh dsk kre mecp ketelt order tg d hy memerlk tg ttk, dbdgk metode smpso /8 yg membthk empt
ttk. Dlm pemk byk ps, metode smpso / hy berlk tk jmlh ps gep. Apbl dkehedk jmlh ps gjl, mk dpt dgk metode trpesm. Tetp metode tdk begt bk kre dy keslh yg ckp besr. Utk t ked metode dpt dgbg, yt sejmlh gep ps dgk metode smpso / sedg ps ssy dgk metode smpso /8. Cotoh sol: Deg tr smpso /8 htg I e d. Htg pl tegrl tersebt deg meggk gbg dr metode smpso / d /8, pbl dgk 5 ps deg =,8. Peyeles: Metode smpso /8 deg st ps Itegrl dhtg deg meggk persm (: ( ( ( ( I ( b 8 ( e I ( e, e 8,6667 e 55,7798. Besr keslh dlh: 5,5985 55,7798 t %,76 %. 5,5985 b Apbl dgk 5 ps, mk dt tk kelm ps tersebt dlh: ( = e = (, = e, =,8. (,8 = e,8 =,55 (, = e, =,55. (,6 = e,6 =,95 ( = e = 5,5985. Itegrl tk ps pertm dhtg deg metode smpso / (persm 7: A b 6 ( ( c ( b I,6 6 ( (,55,95,968. Tg ps terkhr dgk tr smpso /8: ( ( ( ( I ( b 8
(,95 (,8 (,55 5,5985 I, 9,8659. 8 Itegrl totl dlh jmlh dr ked hsl dts: I,968 9,8659 5,8687. Keslh terhdp l eksk: 5,5985 5,8687 t 5,5985 %,7 %... lgortm metode tegrs smpso ( desk y=( ( tetk bts bwh ( d bts ts tegrs (b ( tetk jmlh pembg ( htg h=(b-/.. Atr trpezod Itegrs tk poloml orde ke- k berhbg deg tr trpezod tk meghtg tervl tr d bh ttk. Utk bts ts tegrs yg ekvle deg s =, tegrs tk setp segme dlh: ( ( Dm: h =, d = ( +, sehgg: ( ( [ ] ( Gmbr 5.5 lstrs tegrs tr Gbg ls dr tp tervl k mejd:
( Dm h = ( +. Persm dts tdk hrs mempy tervl yg sm. Utk pembg tervl yg sm, persm d ts k dpt mejd: ( Dm : = = h = kostt. Itegrs romberg d kdrt gss.. Metode romberg Metode tegrl romberg ddsrk pd perls ekstrpols rchrdso tk memperoleh l tegrl yg semk bk. Perl dketh bhw setp peerp ekstrpols rchrdso k mekk order glt pd hsl sols sebesr d: Msly, bl (h d (h dhtg deg kdh trpesm yg berorde glt o(h, mk ekstrpols rchrdso meghslk kdh smpso yg berorde o(h, seljty (h d (h dhtg deg kdh smpso, ekstrpols rchrdso meghslk kdh boole, yg berorde lt o(h 6. Tj kembl persm ekstrpols rchrdso : ( ( ( Mslk dlh l tegrl sejt yg dytk sebg : Yg dlm hl, A k = perkr l tegrl deg kdh trpesm d jmlh segme = k. Orde glt k dlh o(h. Sebg cotoh, selg [, b] dbg mejd 6 segme :
Art dr setp k dlh sebg berkt : A = tksr l tegrl ( deg meggk kdh trpesm d pembg derh tegrs mejd = = bh segme A = tksr l tegrl ( deg meggk kdh trpesm d pembg derh tegrs mejd = = bh segme A = tksr l tegrl ( deg meggk kdh trpesm d pembg derh tegrs mejd = = bh segme A 6 = tksr l tegrl ( deg meggk kdh trpesm d pembg derh tegrs mejd = 6 = 6 bh segme Tg k pertm dlksk oleh gmbr sepert d bwh A A A
Gk,,,., k pd persm ekstrpols rchrdso tk medptk rt b, b,., b k yt : Jd l (yg lebh bk sekrg dlh = b k + d h + e h 6 +... Deg orde glt b k dlh o(h. Seljty gk b, b,., b k pd persm ekstrpols rchrdso tk medptk rt c, c,., c k, yt : Jd l (yg lebh bk sekrg dlh = c k + e h 6 +... Deg orde glt c k dlh o(h 6 Seljty gk c, c,., c k pd persm ekstrpols rchrdso tk medptk rt d, d,., d k, yt : Jd l (yg lebh bk sekrg dlh = d k + e h 8 +... Deg orde glt d k dlh o(h 8. Demk setersy. Dr rt tersebt dperoleh tble dsebt tble romberg, sebg berkt :
Cotoh sol :. Htg tegrl deg metode romberg ( = 8. Gk 5 gk d belkg kom. Jwb : Jrk tr ttk : h = (-/ 8 =,5 Tble ttk ttk ddlm selg [, ] deg h =,5:
. Metode kdrtr
D dlm metode trpesm d smpso, gs yg dtegrlk secr merk terdr dr d betk yt tbel dt t gs. Pd metode kdrtr, yg k dbhs dlh metode gss kdrtr, dt yg dberk berp gs. Pd tr trpesm d smpso, tegrl ddsrk pd l-l d jg-jg ps. Sepert pd gmbr 7.9, metode trpesm ddsrk pd ls d bwh grs lrs yg meghbgk l-l dr gs pd jg-jg tervl tegrs. Rms yg dgk tk meghtg ls dlh: I ( b ( ( b (7. deg d b dlh bts tegrs d (b dlh lebr dr tervl tegrs. Kre metode trpesm hrs mell ttk-ttk jg, mk sepert terlht pd gmbr 7.9. Rms trpesm memberk keslh ckp besr. Gmbr 7.9. Betk grk metode trpesm d gss kdrtr D dlm metode gss kdrtr dhtg ls d bwh grs lrs yg meghbgk d ttk sembrg pd krve. Deg meetpk poss dr ked ttk tersebt secr bebs, mk k bs dtetk grs lrs yg dpt meyembgk tr keslh post d egt, sepert pd gmbr 7.9b. Dlm metode trpesm, persm tegrl sepert dberk oleh persm (7. dpt dtls dlm betk: I c ( c ( b (7.5 Deg c dlh kostt. Dr persm tersebt k dcr koese c d c. Sepert hly deg metode trpesm, dlm metode gss kdrtr jg k dcr koese-koese dr persm yg berbetk:
( ( c c I (7.6 Dlm hl vrbel d dlh tdk tetp, d k dcr sepert pd gmbr 7.. Persm (7.6 megdg blg tk dketh, yt c, c,, d, sehgg dperlk persm tk meyelesky. Utk t persm (7.6 dggp hrs memeh tegrl dr empt gs, yt dr l ( =, ( =, ( = d ( =, sehgg tk: ( ( : ( c c d c c (7.7 ( ( : ( c c d c c (7.8 ( ( : ( c c d c c (7.9 ( ( : ( c c d c c (7. Sehgg ddpt sstem persm: c c ; c c ; c c ;. c c Peyeles dr sstem persm dts dlh: c = c = ; = =,577569; = =,577569. Sbstts dr hsl tersebt ke dlm persm (7.6 meghslk: ( ( I (7.
Gmbr 7.. Itegrs gss kdrtr Bts-bts tegrl dlm persm (7.7 hgg persm (7. dlh smp, sehgg lebh memdhk htg d membt rms yg ddpt bs dgk secr mm. Deg melkk trsorms bts-bts tegrs yg l dpt dbh ke dlm betk tersebt. Utk t dggp terdpt hbg tr vrbel br d d vrbel sl secr ler dlm betk: = + d (7. Bl bts bwh dlh =, tk vrbel br bts tersebt dlh d =. Ked l tersebt dsbsttsk ke dlm persm (7., sehgg dperoleh: = + ( (7. D bts br d =, memberk: b = + ( (7. Persm (7. d (7. dpt dselesk secr smlt d hsly dlh: b (7.5 D b (7.6 Sbsttsk persm (7.5 d (7.6 ke persm (7. meghslk: ( d b ( b Deresl dr persm tersebt meghslk: b d d d (7.7 (7.8 Persm (7.7 d persm (7.8 dpt dsbsttsk ke dlm persm yg dtegrlk. Betk rms gss kdrtr tk d ttk dpt dkembgk tk lebh byk ttk, yg secr mm mempy betk: = c ( + c ( + + c ( (7.9 Nl c d tk rms smp deg em ttk dberk dlm tbel 7..
Tbel 7.. Nl c d pd rms gss kdrtr Jmlh ttk 5 6 Cotoh sol: Koese c C =, C =, C =,555555556 C =,888888889 C =,555555556 C =,78585 C =,65555 C =,65555 C =,78585 C =,696885 C =,786867 C =,568888889 C =,786867 C 5 =,696885 C =,79 C =,67657 C =,67995 C =,67995 C 5 =,67657 C 6 =,79 Vrbel X =,577569 X =,577569 X =,77596669 X =, X =,77596669 X =,866 X =,998 X =,998 X =,866 X =,967986 X =,5869 X =, X =,5869 X 5 =,967986 X =,9695 X =,66986 X =,86986 X =,86986 X 5 =,66986 X 6 =,9695 Htg tegrl I e d, deg meggk metode gss kdrtr. Peyeles: Deg meggk persm (7.7 tk = d b = ddpt: ( d b ( b ( (( d Tr dr persm tersebt dlh: d
d = d d Ked betk dts dsbsttsk ke dlm persm sl, sehgg ddpt: e d e (d d d Rs k dr persm dts dpt dgk tk meghtg ls deg metode gss kdrtr, deg memskk l d = =,577569 d l d = =,577569. tk =,577569 ( (,577569 e,6575. tk =, 577569 (,577569 e 6,8997. Ls totl sepert dberk oleh persm (7.: =,6575 + 6,8997 = 5,598. Keslh: 5,5985 5,598 t 5,5985 Cotoh sol: %,8 %. Htg tegrl I e d, deg meggk metode gss kdrtr ttk.tk ttk persm (7.6 mejd: I c ( c ( c ( Sepert terlht dlm tbel 7., tk ttk, koese c d dlh: c =,555555556. X =,77596669. c =,888888889. X =,. c =,555555556. X =,77596669. Dr cotoh sol sebelmy ddpt persm yg telh dkovers dlh: (c e d e (d d d ( tk =,77596669 e,9556. ( tk =, e,778. ( tk =,77596669 e 69,5795. Persm (c mejd: = (,555555556,9556 + (,888888889,778
Keslh: + (,555555556 69,5795 = 5,586. t 5,5985 5,586 5,5985 %, %.
Bb Pembhs Metode grk (oleh re krw. ( (reeres : bk metode merk bb sol o.. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( X F(,6,6,8,6,, -,7, -,5
-,6 -,. Tetk kr kr yt dr Deg meggk metode grk Jwb: Metode grk tebk wl = - tervl =, - - -.8 -.8.8,6,8, -.6 -.6.6,6.6, 96 -, -,,,6,, 56 -. -..... Jd l yg plg medekt dlh Metode bg d (yog tr wrme. Thermstor merpk lt pegkr tempertre Thermlly codctve epoy cotg T plted copper lloy led wres gre thermstor. Hbg tr hmbt R dr thermstor d tempertr dlh.9.77 l( R 8.77568 T l 8 R
DmT dlm kelv d R dlm ohms. Glt thermstor tk lebh dr.c. Utk meemk rge hmbt pd sh 9 C, kt perl memechk persol..9.77 l( R 8.77568 9. 7.5 l R D.9.77 l( R 8.77568 8.99 7.5 l R 8 8 Fd the bsolte reltve ppromte error t the ed o ech terto d the mber o sgct dgts t lest correct t the ed o ech terto. Gk metod bg d dlm mecr kr persm tk mcr hmbt r pd 8.99 C (deg kl ters d cr jmlh dgt yg sgk yg ber pd khr setp ters! Solto.9 8.99 7.5 Kt dptk ( R.77 l( R 8.77568 Jk kt bersms R, R Mk ( R ( ( R.77.56 ( 5.77 5.756 Oleh kre t R R Iterto R.77 l( R 8.77568 l 8 l( 8.77568 l( 8.77568 lr.9775 8 8 l l.9775.9775 5 5.56.756 m R R 5 8 R
( R m (5.77.655 5 l(5 8.77568 8 l5.9775 5 5 R R 5.56.655 m Kre kr-kry berd dtr bts bwh d tsy mejd 5, R R R m d R, t dtr 5 d. Sehgg belm dpt dtetk meggt merpk ters pertm. Iterto R R Rm 5 5 R 5 m.77.599 6 l 8 5 8.77568 l5 5 6 R R 5 5.655.599 m.9775 Kre kr-kry berd dtr R m d R, t dtr 5 d5. Sehgg bts bwh d tsy mejd, R 5, R 5 ew old Rm Rm ew Rm 5 5 5 5.66% Iterto R R Rm 55 875 R 875 m.77. 6 l 8 875 8.77568 l875.9775 5 6 R R 5 875.65.98 m Kre kr-kry berd dtr R d R, t dtr 875 d 5. m
Sehgg bts bwh d tsy mejd, R 875, R 5 ew old Rm Rm ew Rm 875 5 875.96% Tble Iterto 5 6 7 8 9 R 5 5 875 6 6 6 6 7 7 R 5 5 5 56 9 86 86 8 R % m 5 5 875 6 56 9 86 7 8 77 ---------- 5.66.96.5.759.5757.79.896.796. R m 5.655 6.599..7 6.59 7 6.97 7.79 8. 8.7.57 6 7 8 8 Pd ters ke-,.% Oleh kre t jmlh gk sgk yg ber dberk oleh l m terbesr dlh m.5 m..5 m.86 log.86 m m log.86. 87 So m The mber o sgct dgts t lest correct the estmted root 77 s.
. Bol megpg memlk gy grk bm sebesr.6 d rds bol dlh 5.5cm. D k dmt tk meemk kedlm yg bol teredm st megpg d r. Persm tk kedlm bol yg teredm oleh r dlh.65.99 Gk metode bg d tk meemk kr persm tk meemk kedlm yg bol teredm r. Lkklh tg ters tk memperkrk kr persm d ts. Cr mtlk keslh perkr relt pd khr setp ters, d jmlh gk sgk yg ber pd khr ters ketg. Solto Fgre lotg bol problem. Bol megpg dtr d R, Dm R rds o the bll, Jk R (.55.,. ( ( (.65(.99.99 ( (. (..65 (..99 Oleh kre t, ( ( ( (. (.99 (.66.66 Setdky d st kr yg berd d, dtr d.. Iterto m..55.55 5.55.65.55.99 6.655 m
( ( m ( (.55 Kre l kr-kry berd pd bts bwh d tsy mejd.55,. Iterto The estmte o the root s m.55..85 ( (.85 (.85 m.99 6.655 m d.65(.85, yk dtr.55 d.. Sehgg.99.6 5.55.85 6.655.6 m Kre l kr-kry berd pd d Sehgg bts bwh d tsy mejd.55,.85 ew old m m ew m.85.55.85.% Iterto m.55.85.6875 ( (.6875 (.6875 m ( ( m.65(.6875 m, yk dtr.55 d.85..99 5 (.55 (.6875 (6.655 ( 5.56 Kre l kr-kry berd pd d Sehgg bts bwh d tsy mejd.55,.6875 5 5.56 5 m, yk dtr.55 d.6875. ew old m m ew m.6875.85.6875 % Iterto m % ( m 5...55 ---------- 6.655
.55..85..6.55.85.6875. 5 5.56.55.6875.688. 6.8 5.688.6875.65 5.6 5.59 6.688.65.659.7 5.8 7.688.659.67.7 6.76 8.688.67.6.6897 7 6.97 9.6.67.65.6 6.65.6.65.6.7 7.768 Pd khr ters.7% Jmlh gk sgk yg ber dr kr dr ters terkhr dlh m.5 m.7.5. m log(. m m log(..6 Sehgg m Jmlh gk sgk yg ber dr kr dr ters dr kr. 6 pd khr ters dlh. Metode poss pls (bdl hrrs hdyt. Dketh persm (. Asmsk l = d l =,5 sert bts tolers dlh %. (reeres : bk metode merk bb 5 sol o. 5. Iters X = X =,5 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Mk br = r
Iters X = X =, ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Mk br = r Iters X = X =,656 ( ( ( ( ( ( ( ( ( ( ( ( ( ( Jd l kr dr gs tersebt dlh,666 deg keslh,6% Iters st ttk sederh (re krw e -. Dketh - Asmsk l,8 d tolersdlh %
(reeres : bk metode merk bb 5 sol o.5.6 g e e,8 e e,899,899 e e,8 r br r lm,8,899 % %,%,8 r br,8 e e,886 r br r lm,886,8 % % 8,5%,886 r br,886 e e,86 r br r lm,86,886 % % 7,%,86 r br,86 5 e e,8768 r br r lm,8768,86 % % 5,9%,8768 r br,8768 6 e e,8 r br r lm,8,8768 % % 5,6%,8 r br,8 7 e e,87 r br r lm,87,8 % %,8%,87 r br,87 8 e e,86
r br r lm,86,87 % %,%,86 r br,86 9 e e,87 r br r lm,87,86 % %,5%,87 r br,87 e e,8 r br r lm,8,87 % %,%,8 r br,8 e e,86 r br r lm,86,8 % %,7%,86 r br,86 e e,86 r br r lm,86,86 % %,6%,86 r br,86 e e,858 r br r lm,858,86 % %,%,858 r br,858 e e,88 r br r lm,88,858 % % %,88 r br Jd, l kry dlh,88 deg keslh sebesr % Metode ewto rphso (bdl hrrs hdyt ( (reeres : bk metode merc bb 5 hl 66 cotoh sol 5.
Bts tolers F( = + Iters ( '( Iters Nl = (( ( (,5.5,5 % %,5 Iters Nl =,5 ((,5 (,5,5 (,5,5,5,5 5,5 (,5 %,857%,5 Iters Nl =,5 ((,5 (,5,5 (,5,5,5,
,69756,69756 (,5 %,9%,69756 Iters Nl =,69756 ((,69756 (,69756,69756 (,69756,6975 6,5956,9,,595 (,69756 %,6%,595 Iters Tbel ters X Ε tolers,5 %,5,85 %, 69756,9 %,69756.6 % Akr - kr yt dr ( deg bts tolers % dlh,69756 pd ters ke.. Kt k membt sebh rk bk berod deg tgg dr 8½" hgg " d pjg 9". Ky yg dgk memlk l yog s modls.667ms, ketebl /8" d lebr ". Persm tk mecr l deleks vertcl dlh 8 5 6 v(.9.5.667. 857
Dm dlh st poss yg terletk pd blok. D tk mecr deleks dv mksmm kt hrs meemk letk ( d melkk perk d persm d kl Books Bookshel Fgre loded bookshel. Persm tk poss pd l deleksy mksmm.67665 8.6689 5.78.857 Gk metod ewto-rphso tk meemk kr persm tk meetk poss yg berdeleks mksmm (lkk ters sebyk kl sert tetk l hmpr pd khr setp ters. Solto Iterto 8 5.67665.6689.78.857 8 5.766.867.596, mk l Jk bersms l. 8 5.67665 (.6689 (.78 (.857 8 5.766 (.867 (.596 ( 8.956.79.99.9
.9.9.8% Iterto (.9.596 (.9.867 (.9.766.857 (.9.78 (.9.6689 (.9.67665.9 5 8 5 8.97 6.989.9.69.9.57.57.9.57.86% Iterto (.57.596 (.57.867 (.57.766.857 (.57.78 (.57.6689 (.57.67665.57 5 8 5 8 9.9.778.57 6.75.57.57.57.57.57 5.677
Metode sect (re krw. Tetk kr-kr yt dr persm berkt: F( =.7 + 6..97 + 9. Jwb: Dm tebk wl r- =.5, r =, ε =. (r (r r ε r r r r % (r (r Iters ( r (.5.7.5 9.99 ( (.7 r 9.6 6. 6..5.97.5 9..97 9. 9.6 (.5 r ε.67.6 9.6 9.99 9.6 (.5 9.6 9.99 9.658 8.87.6.67 Iters ( ( 9.6 r ( (.67 r.7.67 6..67.97.67 9. 97.78 97.78 (.67 r.67 ε.87.67.966 97.78 9.6 59.678.5.67.966.87
Iters ( (.67 r 97.78 ( (.87 r.7.87 6..87.97.87 9. 9.58 9.58 (.87.67 r.87 ε.6.87.59 9.58 97.78.586.87 8.7.87.59.6 Iters ( (.87 r 9.58 ( (.6.7.6 6..6.97.6 9. r 8.97 8.97 (.6.87 r.6 ε.6.6.65 8.97 9.58 9.66.6.5.6.65.6 Iters 5 ( (.6 r ( (.6 r 8.985 8.97.7.6 6..6.97.6 9. 8.985 (.6.6 r.6 ε 5.79.6.5 8.985 8.97.68.6.5755.6.5.79 Iters 6 ( (.6 r 8.985
( (.79 r.7.79 6..79.97.79 9..8.8 (.79.6 r.79 ε 6.568.79.5.8 8.985.98.79.8.79.5.568 Kre hrg keslh (ε y telh kecl dr yg dtetk mk ters berhet, d ddptk kry: X r,568. Ad memlk sebh tgk peymp myk. Dmeter tgk 6 t. Ad k dmt tk dmecr tgg tgk h,dpstck 8 t k bsh jk tgk ds myk sebyk t. Dpstck Sphercl Storge Tk r h Fgre sphercl storge tk problem. Persm tk tgg h zt cr dlm tgk jk volm d jr-jr dketh h h 9h.897 Gk metod sect tk meetk kr persm tk mecr tgg h dr dpstcks yg bsh oleh myk. Lkk ters sebyk kl d tetk l glt reltve dsetp ters Solto Jk kt megsmsk l tebk wl tk h dlh h. 5 d h.
Iterto h h h h h h h h h 9h.897 h h h 9h.897 h 9h.897 9.897.5 9.897.5 9.5.6.897 h h h.6.6 55.% Iterto h h h h h h h h h 9h.897 h h h 9h.897 h 9h.897.6 9.6.897.6.6.6785.6 9.6.897 9.897 h h h.6785.6.6785.% Iterto h h h h h h h h h 9h.897 h h h 9h.897 h 9h.897.6785 9.6785.897.6785.6 9.6785.897.6 9.6.6785.6785.897
. 67759 h h h.67759.6785.67759.8768% Elms gss (yog tr wrme. Kecept roket terhdp wkt dtjkk pd tbel Kecept dgmbrk dlm persm v t t t, 5 t Tme, t (s Velocty, v (m/s 5 6.8 8 77. 79. 5 5 6.8 6 8 77. 79. Tetk l,, deg elms gss Solto Kre d persm mk dperlk lgkh. Lgkh Bg brs deg 5 d dkl 6 sehgg brs deg 6/5.56. 5 5 6.8. 56 6.8.56 7.8 Dkrg deg brs 6 8 77. 6.8.56 7.8.8.56 96.8 Sehgg ddptk
5 5 6.8.8.56 96.8 79. Bg brs deg 5 d dkl sehgg brs deg /5 5.76 5 5 6.8 5. 76 8.8 5.76 65.68 Dkrg deg brs 79. 8.8 5.76 65.68 6.8.76 5.968 Sehgg ddptk 5 Lgkh 5.8 6.8.56.76 6.8 96.8 5.968 Sekrg kt bg brs.8 d dkl deg 6.8 sehgg dsmplk brs dkl 6.8/.8.5..8.56 96.8. 5 6.8 5.6 6.78 Dkrg deg brs 6.8.76 5.968 6.8 5.6 6.78.7 Sehgg ddptk 5 5.8.56.7.7.76. 76. 7.857 Sbsttsk pd persm.8.56 96.8 96.8.56.8.76 6.8 96.8.76
96.8.56.857.8 9.695 Sbsttsk d pd persm 5 5 6.8 6.8 5 5 6.8 59.695.857 5.97 Hsl khry mejd.97 9.695.857 Sehgg, v t t t.97 t 9.695 t.857, 5 t Metode gss jord (bdl hrrs hdyt. Selesk sstem persm berkt deg metode gss-jord: + y z = 5 + 7y z = y + 5z = (c. (c.b (c.c Peyeles: Sstem persm dts dtls dlm betk mtrks sebg berkt: 7 5 y 5 z (c, 7,,6666 y 5 z (c
6,6668,6,6666 5,6666,6666,6668 5,6668,, z y (c 6,6668,59,6666 5,6666,6666,9,, z y (c5,9,59,88,88,9,5 z y (c6,655,59,88,9,5 z y (c7,655,,56 z y Dr sstem persm dts, ddpt l, y d z berkt : =,56; y =, d z =,655. Metode gss sedel (yog tr wrme. Fd the solto to the ollowg system o eqtos sg the gss-sedel method. 5 8 5 76 7 Use
As the tl gess d codct two tertos. Solto The coecet mtr A 5 7 5 Is dgolly domt s 5 8 5 5 7 Ad the eqlty s strctly greter th or t lest oe row. Hece, the solto shold coverge sg the gss-sedel method. Rewrtg the eqtos, we get 5 8 5 76 7 Assmg tl gess o Iterto # 5.5
.5 8 5.9 7.9 76.5.9 The bsolte reltve ppromte error t the ed o the rst terto s.5.5.%.9.9.%.9.9 67.66% The mmm bsolte reltve ppromte error s.% Iterto #.679.9 5.9 8.679.9 5.75 7.75 76.679.88
At the ed o secod terto, the bsolte reltve ppromte error s.679.5.679.6%.75.9.75.889%.88.9.88 8.87% The mmm bsolte reltve ppromte error s.6%. Ths s greter th the vle o.% we obted the rst terto. Is the solto dvergg? No, s yo codct more tertos, the solto coverges s ollows. Iterto % % %.5..9..9 67.66.679.6.75.889.88 8.87.775 8.6.6 7.8.978.6.9675.56.8.996.997.6577 5.9977.59..899..78 6.9999.77..856.. Ths s close to the ect solto vector o. Dketh persm :
Jwb : Asmsk l dlh ol mk: Msk l dts deg sms ( dlh ol ( ( Iters ( ( ( ( ( ( ( ( Metode crot (Are Krw. Dketh persm d betk persm mtrk:
9 8 6 9 8 6 Dketh = Jwb :.,, ;,,,,,, L L d j tk U j deg L U L L L 7 7 (. U L U L L L U L U U L L
(, 7 5,, D D D D D D D D D Metode dooltle (Yog Tr Wrme. Lkk opers ktorss tp pekr brs A = 5 8 6 6 8 5 6 6 6 6 7 Peetp ts brs dr d kolom dr l meghslk: 5 7 6 6 : : 8 6 5 : 8 6 6 :............ :... : Peetp brs dr d kolom dr l : 5 6 5 6 8 8 6 6 7 :... :............ : : : : peetp brs dr d kolom dr l membw kepd hsl dbwh :
5 6 7 8 8 7 6 7 6 7 : :...... :......... : : / : / peetp brs dr, kolom dr l : 5 6 7 8 8 7 75 7 7 6 7 8 7 : : :......... :...... / : / / : / Peetp eleme terkhr dr (ses rms member : 5 / 8/ 7 6 / 7 7 75/ 7 / 8 8 7 6 l. Jd hsl khr dlh L = 8/ 7 6 / 7 / 8 8, = / 7 7 75/ 7 6. Sekrg jk ss k persm ler = b dlh b = [ 5] t, mk pd khry dperoleh sols = [.689.78-6.6 -.68 -.9] t.
Metod cholesky (Abdl Hrrs Hdyt. Selesk persm ler berkt : j =, d = k = mk,... j = ; t = j m = U = k = -, ˆ K d j = ; t = j m = = k = -, j = ; t = j m = = k = -,, ˆ k,...,; y Q y m k k k q q q q ˆ d ˆ d ˆ d ˆ K d ˆ K d ˆ ˆ ˆ d d d d ˆ ˆ ˆ y Q y y Q y y Q y Q y
. Dketh sebh mtrk : [ ] [ ] (reeres : bety prstw, mklh metode cholesky. 8 Lgkh yg hrs dlkk :. Cr l [] Utk,, berl ol ( kre > j [ ] [ ]. Cr l [ t ] Megbh brs mejd kolom d seblky [ ] [ ]. Pembkt Bktk bhw [] = [ t ][] Progrm mtlb % dekomposs metode cholesky % tk mtrk smetr d post
% cr pggl: c = cholesky( % pt: = mtrk smetr d post % otpt: c = mtrk segtg ts dm: = c*b A=[ -; 5 7; - 7 7] [m,] = sze(; I m~=, error('mtrk hrs bjr sgkr'; Ed B=chol ( C=trspose (b A=c*b Metode regres ler (Are Krw. X Y 6 8 8 8 6 6 8 8 8 (reeres : metode merk bk jr l
Progrm mtlb T=[ 6 8 6 8]; V=[ 8 8 6 8 8]; P=polyt(t,v,; X=lspce(,8,; y=polyvl(p,; Plot(,y,'k-',t,v,'k*' P P = -.6 -.656 8.85 5 5 5 5 5 5 Metode regres poloml (Yog tr wrme Tble tbel tgg k-k berdsrkperbedmr t (yrs 5. 8 6 8 H ( 6. 5 6 69. 7 Perkrktggk-kketkbermr thdegmemk tble pertmbhk-kdpersm
H be ct Solto Persm yg dgkdlh H be ct Dmkostt,b, c dlhkrpersm o-ler 6 6 6 e ct e ct e ct e bh bte H ct e ct ct b e e b bh ct e b ct H ct e b H ct (,b,c Ktmemerlktebkwltkmedptklkry. Mslyktmeggk(,, (, 6 d (8, 7 c be ( 6 c be 7 c be ( (8 Akr, b d c drpersmdts 7.55 b.7767 c.977 Jkmeggkmetode ewto-rphso 7. b.8 c.75
Jkdmskkkedlmpersm h 7. H.75 t.8 e Tggk-kketk th H 7..75.8e 7" ( Metode terpols ler (Abdl Hrrs Hdyt. tksrlh logrtm (l deg meggk terpols ler Jwb: Dk: kompts. L = d l 6 =.797595 kopts. L = d l =.869.797595 ( (.5859 6.869. 698 Jd deg tervl yg lebh pedek megrg % keslh.. Cr l y tk ttk =. yg berd d tr ttk (,.5 d (,.5 (reeres : bh jr ts, metode merk. -terpols.pd
Ttk p (.,.5 Metode kdrtk (Yog Tr Wrme. Htg tksr y tk = deg meggk terpols kdrtk tk dt : (,, (,.869, (6,.79759 (reeres: dktt klh, versts krste sty wc. Cotoh 5. hl 9 Koese-koese dr persm terpols kdrtk Sehgg terpols kdrtk tk dt yg dberk yt Kre t, l y tk = yt Dk:. Htglh l l. 6.869. 797595 Jwb: b b b.869.698.797595.869.698 6.5876 6
.698.5876 Dm =. 56586 Atr trpesm d tr smpso (Are Krw. Cotoh sol: ( = + deg bts bwh d bts ts 5 deg sb tervl t segme ( = 5. A. Tetk peyelesy deg metode trpesm B. Tetk peyelesy deg metode smpso / Jwb : A. Deg metode trpesm F ( = F ( = F ( = F ( = F ( = F (5 = [ ( ( ( ( ( ( ( ] [ ( ] [ ] B. Deg metode smpso / F ( = F ( = F ( = F ( = F ( = F (5 = [ ( ( ( ( ( ( ( ( ]
[ ( ( ] [ ]. Cotoh sol : ( deg bts bwh d bts ts,5 deg sb tervl t Segme ( = 6. A. Tetk peyelesy deg metode trpesm B. Tetk peyelesy deg metode smpso / Jwb : A. Deg metode trpesm X = ( = ( ( = X = (,5 = ( ( =,6565 X = (,5 = ( ( =,65 X = (,75 = ( ( =,875 X = ( = ( ( = X 5 = (,5 = ( ( =,5 X 6 = (,5 = ( ( =,875 [ ( ( ( ( ( ( ( ( ] [ ( ] [ ( ] ( B. Deg metode smpso / X = ( = ( ( = X = (,5 = ( ( =,6565 X = (,5 = ( ( =,65 X = (,75 = ( ( =,875 X = ( = ( ( = X 5 = (,5 = ( ( =,5 X 6 = (,5 = ( ( =,875
[ ( ( ( ( ( ( ( ( ( ] [ ( ( ] [ ] Itegrs romberg (Abdl Hrrs Hdyt. Jrk vertcl dlm meter yg dtemph roket dr t 8 hgg t seko dperoleh oleh persm l 9.8t dt t 8 Gk metod romberg s tk meemk jrk tersebt. Gk hsl dr tbel. Solto Dr tble ddptk I 868 I I I,,,, 66 7 Utk medptk l ekstrpols orde, I, I, I, I, 66868 66 65 I, I, I, I, 66 6 I, I, I, I, 7 7 Utk medptk l ekstrpols orde, I, I, I, I, 5 665 6 5
I, 6 I, I, I, 5 66 6 5 6 Utk medptk l ekstrpols orde, I, I, I, I, 6 66 6 6 6m Tble -segmet -segmet -segmet 8-segmet 868 66 7 Orde Orde Orde 65 6 6 6 6 6. Jrk vertcl dlm meter oleh roket dr t 8 hggt seko dytk dlm persm : l 9.8t dt t 8 Gk metod romberg tk jrk. Gk hsl dr metod trpezm dr tble Solto A I 66 m I m Deg meggk orml rchrdso I I TV I Deg, I I TV I
66 6m 8 9.8 l dt t t 6m Vle Tre Error Tre t 6 6 6.9%
Dtr pstk Kreyszg, E., Advced Egeerg Mthemtcs, Joh Wley & So, Ic, New Jersey 6. Hom, J.D., Nmerkl Methods or Egeers d Scetsts, Mrcel Dekker, Ic. New York, 99. Srbkt, I., Metode Nmerk, Dktt Klh, ITS ---, Metode Nmerk, Ac Klh, Poltekk, ITS Nm.mthcollege.cgm