Modifikasi Metode Newton-Steffensen Tiga Langkah Menggunakan Interpolasi Kuadratik

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1 Smir Nsiol Tkologi Iormsi, Komuiksi d Idusri SNTIKI ISSN : Pkbru, Novmbr 0 Modiiksi Mod Nwo-Ss Tig Lgkh Mgguk Irpolsi Kudrik Wroo, Ek Jumii, Progrm Sudi Mmik, UIN Sul Sri Ksim Riu Jl. Subrs km, Pkbru 89 -mil: wroo@ui-susk..id Absrk Mod Nwo-Ss mrupk mod irsi g diguk uuk mlsik prsm olir g dihsilk dri modiiksi mod Ss. Pd mklh ii, pulis mgmbgk mod Nwo-Ss g diulis olh Shrm [0] dg mgguk irpolsi kudrik. Brdsrk hsil kji diprolh prsm irsi bru dg ord kovrgsi m g mlibk lim vlusi ugsi dg idks isisi,09. Sli iu, simulsi umrik dilkuk rhdp bbrp ugsi dg ili bk wl g brbd d diprolh bhw prorm modiiksi mod g bru lbih bik dibdigk dg mod Nwo, mod Ss d mod Nwo-Ss. K kui: idks isisi, mod Nwo-Ss, ord kovrgsi, prsm olir Absr Th Nwo-Ss s mhod is iriv mhod h usig or solvig olir quios rsulig rom h modiiio o h Ss s mhod. I his ppr, h uhor dvlopd h Nwo-Ss s mhod wri b Shrm [0] usig b h qudri irpolio. Bsd o h sud rsuls obid h w irio quio wih si-ordr ovrg h ivolvig iv vluio uios wih ii id,09. Bsids, umril simulios prormd o som uios wih dir iiil guss vlu d obid h prorm o h w mhod is br h h Nwo s mhod, Ss s mhod d Nwo-Ss s mhod. Kwords: ii id, Nwo-Ss mhod, ord o ovrg, olir quio. Pdhulu Mod Nwo mrupk slh su mod irsi dg ord kovrgsi kudrik uuk muk kr-kr prsm olir dg mgguk bk wl g dirumusk olh Sli iu, rdp mod irsi li g mmiliki ord kovrgsi kudrik g diprolh dg mggik uru prm Nwo dg buk +, shigg rumus bru mod irsi rsbu dlh dg sisi idks sm dg mod Nwo, iu sbsr, Uuk migkk ord kovrgsi, bbrp plii mgmbgk suu mod irsi bru du lgkh uuk mghsilk ord kovrgsi kubik dg mgguk brbgi pdk, spri: bd higg rpus [, ], bd mju dg prmr uggl [, ], polomil kudrik d kubik [], irpolsi lir [9], bd mju dg prmr ugsi [0] d koisi k u []. Pd mklh ii, k dikmbgk mod irsi du lgkh g lh dikji olh Shrm [0] dg ord kovrgsi kubik dlm buk, 09

2 Smir Nsiol Tkologi Iormsi, Komuiksi d Idusri SNTIKI ISSN : Pkbru, Novmbr 0 0 dg Prsm mrupk modiiksi mod Ss, d lbih dikl dg m mod Nwo-Ss g mlibk vlusi ugsi, iu d d, su vlusi uru prm ugsi, iu, shigg mod rsbu mmpui idks sisi sbsr /, Uuk migkk ord kovrgsi mod Nwo-Ss, mk dikoruksi mod irsi ig lgkh dg mmbhk lgkh kig dlm buk Nwo g dibrik pd prsm. Sli iu, uuk migkk isisi mod irsi, mk prlu mrduksi buk uru prm pd lgkh kig dg mgguk pdk irpolsi kudrik sbgim g dilkuk olh Chu [] d Kou [8].. Mod Pdg kmbli prsm sbgi briku d dg mgpoksimsik uru prm pd prsm diprolh, dg dlh ugsi sbrg. Slju subsiusi prsm k Prsm, d dg mdrhk prsm rsbu diprolh Jik mk prsm mjdi 8 u 9 dg Prsm 9 mrupk mod Nwo-Ss dg ord kovrgsi kubik g mlibk du vlusi ugsi, iu d, d su vlusi uru prm ugsi, iu. Slju, briku ii k dibrik orm ord kovrgsi dri mod Nwo- Ss. Torm : Mislk D : mrupk ugsi rdirsil sprlu dg kr sdrh, 0 dg D d o ukup dk k mk prsm

3 Smir Nsiol Tkologi Iormsi, Komuiksi d Idusri SNTIKI ISSN : Pkbru, Novmbr 0 mod Nwo-Ss kovrg k dg ord kovrgsi ig g mmuhi prsm gl : 0 dg d! j j. Buki : Adik rror mrupk irsi k-, dg, mk dg mgguk kspsi dr Tlor uuk diskir didpk : lh kr, 0 d, d slju dg msubsiusik buk = + k prsm diprolh : d sdgk uru prm prsm rhdp dibrik olh Brdsrk Prsm d, mk pmbgi dg mmbrik : d dg msubsiusik Prsm k Prsm mk didpk : Slju dg mgguk dr Tlor, mk kspsi di skir dibrik olh!! lh kr 0 d mk : 8 Slju, dg mgguk prsm d 8, mk buk pgurg dg diprolh : 9 shigg prkli dg prsm 9 mmbrik 0 Slju bgik Prsm dg Prsm 0 diprolh : Slju Prsm disubiusik k Prsm 9 diprolh lh kr d, mk prsm mjdi Prsm mrupk ord kovrgsi dri mod irsi g dibrik pd prsm 9.

4 Smir Nsiol Tkologi Iormsi, Komuiksi d Idusri SNTIKI ISSN : Pkbru, Novmbr 0. Hsil d Alisis.. Prsm Irsi d rd Kovrgsi Pdg kmbli prsm mod irsi Nwo-Ss pd prsm g dikmbgk olh Shrm 00, sbgi briku dg sbgim didiisik pd prsm. Slju, skm di s k diubh mjdi skm ig lgkh dg mdiisik kmbli prsm Nwo pd lgkh kig dlm buk dg sbgim didiisik pd prsm. Mrujuk kpd kik g diguk olh Chu [] d Kou [8], buk pd prsm diproksimsi dg mgguk irpolsi kudrik h. Jik h mgirpolsi iik di, d, sr dg mgsumsik = h, mk diprolh buk irpolsi kudrik dlm buk, Slju gik pd prsm dg mk diprolh h lh kr 8 d 9 mk dg msubsiusik buk d msig-msig pd prsm 8 d 9 k prsm d dg mdrhk prsm, mk diprolh 0 lh kr mk prsm 0 dp diubh kmbli mjdi Slju subiusik prsm k prsm shigg diprolh mod irsi Nwo-Ss bru dg buk dg d didiisik pd prsm d.

5 Smir Nsiol Tkologi Iormsi, Komuiksi d Idusri SNTIKI ISSN : Pkbru, Novmbr 0 Prsm mrupk modiiksi mod Nwo-Ss ig lgkh mgguk irpolsi kudrik g mmiliki lim vlusi ugsi iu,,, d. lh kr prsm mmu prmr, mk dg mggi ili k muul bbrp mod irsi bru. Uuk, 0, ] [ d =, Slju, uuk muk ord kovrgsi prsm dilkuk dg mgguk dr Tlor uuk mgkspsi ugsi g rlib pd sip irsi. Uuk iu, briku ii dibrik orm ord kovrgsi prsm. Torm : Mislk I dlh kr dri ugsi R I : uuk suu irvl rbuk I. Jik 0 dlh ili rk wl g ukup dk k mk mod irsi pd prsm. mmiliki prsm gl: 0 9 Buki : Mislk I dg dlh kr dri mk 0 d 0 dg mgguk dr Tlor diprolh d uru prm dibrik 8 Brdsrk prsm mk kudr dlh 9 Slju prsm dibgi dg prsm 8 diprolh 0 Subiusik prsm 0 k prsm d dg mgguk prsm diprolh Brdsrk prsm d dg mgguk kspsi dr Tlor rhdp diskir, mk diprolh d uru prm dlh Slju dg mgguk prsm, 8 d, mk prkli dg dibrik olh

6 Smir Nsiol Tkologi Iormsi, Komuiksi d Idusri SNTIKI ISSN : Pkbru, Novmbr 0 Prsm diguk uuk mghiug rus k pd prsm, diprolh Subiusik prsm k prsm, d mgguk mk diprolh shigg dg mgguk kspsi diskir mk diprolh Slju dg mgguk prsm, 8, d, mk diprolh bruru-uru = shigg diprolh 9 8 Bgik prsm 8 dg prsm diprolh 0 9 Dg msubiusik prsm k prsm d dg mgguk buk, mk diprolh 0 9 lh kr + = + +, mk prsm mjdi 0 9. Simulsi Numrik Pd subbb ii, k diujukk porm mod irsi pd prsm uuk mlsik prsm oliir. lh kr iu, k dibrik simulsi umrik mgguk prgk luk Mpl dg klii 800 digi d kriri pghi kompusi jik mmuhi kodisi briku: g bruju uuk mmbdigk jumlh irsi bbrp mod iri dlm mghmpiri kr prsm olir d dg mgmbil ili bk wl 0 sdk mugki dg kr prsm. Adpu ugsi g k diguk dlh sbgi briku:, 98,99098 si,,098

7 Smir Nsiol Tkologi Iormsi, Komuiksi d Idusri SNTIKI ISSN : Pkbru, Novmbr 0,, , si l, 0, os, 0,908 0 Sli mgguk kspsi dr Tlor, ord kovrgsi mod irsi dp diuk dg mgguk logrim prbdig gl g bis disbu CC ompuiol o ordr ovrg, l l = l l Trki dg CC uuk prsm diprlihk pd Tbl, d g msigmsig uuk ili = -, = 0 d =. Tbl Jumlh irsi, gl mulk d CC prsm dg N CC , , , , , , , , , , Brdsrk Tbl diprolh jumlh irsi d ili CC g mujukk bhw prsm uuk mmiliki ord kovrgsi m. Briku, k dilkuk simulsi umrik prsm uuk 0. Tbl Jumlh irsi, gl mulk d CC prsm dg 0 N CC 0, , , , , , , , , , , , Tbl mmprlihk bk irsi, gl mulk d ord kovrgsi g dihiug sr kompusi. Brdsrk Tbl, dp dilih bhw jumlh irsi prsm uuk ugsi-ugsi g dibrik dlh, d CC prsm dlh m. Hsil CC sr kompusi mguk hsil ord kovrgsi g lh diprolh sr liik. Slju dilkuk simulsi umrik prsm uuk.

8 Smir Nsiol Tkologi Iormsi, Komuiksi d Idusri SNTIKI ISSN : Pkbru, Novmbr 0 Tbl Jumlh irsi, gl mulk d CC prsm dg N CC 0, , , , , , , , , , , Brdsrk Tbl dp dikhui jumlh irsi d ili CC g mujukk bhw prsm irsi uuk mmiliki ord kovrgsi m. Slju, k dilkuk prbdig jumlh irsi r prsm irsi, d dg bbrp mod irsi li, spri: mod Nwo dg ord kovrgsi du NM, mod Ss dg ord kovrgsi du SM, mod Nwo-Ss dg ord kovrgsi ig NSM, modiiksi mod Nwo- Ss, dg NSMM, = 0 NSMM, d NSMM. Briku ii dlh bl prbdig jumlh irsi dri bbrp mod rsbu. Tbl Prbdig Jumlh Irsi uuk Bbrp Mod Irsi Jumlh Irsi NSMM NSMM NSMM 0 NM SM NSM Div Tbl mggmbrk prbdig jumlh irsi dri brbgi mod sbgm lh disbuk sblum dg mgguk bbrp ugsi d ili wl g brbd.. Brdsrk Tbl dp disimpulk bhw jumlh irsi pd modiiksi mod Nwo-Ss NSMM pd prsm uuk, 0, d lbih sdiki dibdigk dg mod Nwo NM, mod Ss SM, d mod Nwo-Ss NSM. Shigg dp dikk bhw NSMM uuk, 0, d mmiliki ord kovrgsi g lbih iggi dri mod li. Hl ii mujukk bhw mod NSMM lbih bik dibdigk mod-mod li.. Ksimpul Pd mklh ii diprolh sbuh mod irsi bru dg ord kovrgsi m, bik hsil sr liik mupu kompusi CC uuk smbrg ili prmr, d mlibk lim vlusi ugsi, iu,,, d, shigg diprolh idks isisi sbsr,09. Jik dibdigk dg mod Nwo g idks isi sbsr,, mk mod irsi prsm lbih isi dibdigk mod Nwo.

9 Smir Nsiol Tkologi Iormsi, Komuiksi d Idusri SNTIKI ISSN : Pkbru, Novmbr 0 Rrsi [] Chu, C., Som Improvms o Jrr s Mhod wih Sih-ordr Covrg, Applid Mhmis d Compuio. 00, 90: [] Dgh, M. d Hjri, M., Som Driviv Fr Qudri d Cubi Covrg Iriv Formuls or Solvig Nolir Equios, Compuiol d Applid Mhmis, 00, 9: 9 0. [] Epprso, J. F., A Iroduio o Numril Mhods d Alsis, Uid Ss o Amri: Wil, 0, [] Hi, M. A., Solvig Nolir Equios Usig Ss-Tp Mhods wih piml rdr o Covrg, Plsi Jourl o Mhmis, 0, : 9. [] Hi, M. A d Bhg, M. S. M., Solvig Nosmooh Equios Usig Fmil o Driviv-Fr piml Mhods, Jourl o h Egpi Mhmil Soi, 0, : 8. [] Hjjh, A., Imr, M. d Gml, M.D.H., A Two-sp Iriv Mhod Fr rom Driviv or Solvig Nolir Equios, Applid Mhmil Sis, 0, 8: [] Jiswl, J.P., 0, A Nw Third-ordr Driviv Fr Mhod or Solvig Noliir Equios, Uivrsl Jourl o Applid Mhmis, 0,,. [8] Kou, J. d Yii Li, 00, A Improvm o h Jrr Mhod, Applid Mhmis d Compuio, No. 89, Hl [9] Liu, Z d Zhg, Q., A -sp Ss-p Mhod wih Supr-ubi Covrg or Solvig Nolir Equios, Prodi Compur Si, 0, 9: [0] Shrm, J.R, A Composi Third rdr Nwo-Ss Mhod or Solvig Noliir Equios, Applid Mhmis d Compuio, 00, 9 :. [] Solmi, F d Hossibdi, V., Nw-Third d Sih-rdr Driviv-Fr Thiqus or Nolir Equios, Jourl o Mhmis Rsrh, 0, : 0. [] Trub, J. F., Iriv Mhods or h Soluio o Equios, Pri-Hll, Eglwood Clis, 9.