SIMULTANEOUS LINEAR EQUATIONS

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1 SIMULTANEOUS LINEAR EQUATIONS T O P I C : G A U S S - S E I D E L M E T H O D /9/9

2 GAUSS-SEIDEL METHOD Prosedur dsr: Adlh metode ITERASI - Meyelesik tip persm liier secr ljbr utuk i - Membut ili sumsi solusi - Selesik utuk tip i d ulgi - Guk perkir keslh reltif tip khir itersi utuk megecek pkh error sudh mecpi gk tolersi.

3 GAUSS-SEIDEL METHOD Kep? Utuk megtsi roud-off error (keslh pembult). Metode elimisi seperti Gussi Elimitio d LU Decompositio(*) rw terhdp keslh pembult.

4 GAUSS-SEIDEL METHOD Sistem persm liier Algorithm... b... b Kit megubh sistem b persm [A]{X}={B} utuk meyelesik deg persm pertm, meyelesik deg persm kedu, d seterusy.

5 GAUSS-SEIDEL METHOD Algorithm Geerl Form of ech equtio c j j j j c j j j j,, j j j j c j j j j c

6 MENJADI: Utuk sistem persm b b b Now we c strt the solutio process by choosig guesses for the s. A simple wy to obti iitil guesses is to ssume tht they re zero. These zeros c be substituted ito equtio to clculte ew =b /.

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8 BATAS AKHIR ITERASI New is substituted to clculte d. The procedure is repeted util the covergece criterio is stisfied: i bru lm i i bru i diperkek t pproimtio error, serig diguk, serigkli disebut sebgi glt bsolut. True error, kurg berrti. diguk Reltive error, dlm prosetse

9 GAUSS-SEIDEL METHOD: EXAMPLE Dikethui sistem persm Iitil Guess: sumsi ili wl, 5

10 GAUSS-SEIDEL METHOD: EXAMPLE Tulis ulg utuk pliksi Guss-Seidel

11 GAUSS-SEIDEL METHOD: EXAMPLE Msukk ili perkir wl utuk selesik i 5 Iitil Guess 6.8 5() (5)

12 GAUSS-SEIDEL METHOD: EXAMPLE Fidig the bsolute reltive pproimte error i ew ew i i old i % % 7.85 At the ed of the first itertio The mimum bsolute reltive pproimte error is 5.47% % 55.6

13 GAUSS-SEIDEL METHOD: EXAMPLE Usig from itertio # Itertio # the vlues of i re foud:

14 GAUSS-SEIDEL METHOD: EXAMPLE Hitug the bsolute reltive pproimte error % % % Akhir itersi kedu Glt bsolut terbesr %

15 GAUSS-SEIDEL METHOD: EXAMPLE Tersuk itersi, kit dptk ili berikut Itertio % % % ! Lho, kok? Error y ggk berkurg?

16 GAUSS-SEIDEL METHOD: PITFALL Slhy dim? ii Cotoh tdi megilustrsik kemugki keslh pd Guss-Siedel method: tidk semu sistem persm k koverge. Is there fi? Oe clss of system of equtios lwys coverges: Oe with digolly domit coefficiet mtri. Digolly domit: [A] i [A] [X] = [C] is digolly domit if: j ji ij Utuk semu i ; DAN ii ij j ji Utuk miiml sebuh i

17 GAUSS-SEIDEL METHOD: PITFALL Digolly domit: Koefisie pd digol hrus sm tu lebih besr dri jumlh semu koefisie pd bris itu, d miiml stu bris hrus memiliki digol yg lebih besr dri jumlh koefisie pd bris itu. Mkh mtriks yg digolly domit? A [ B]

18 GAUSS-SEIDEL METHOD: EXAMPLE Sistem persm liier Deg sumsi ili wl Mtriks Koefisie y dlh A 5 7 Ak koverge kh? 5

19 GAUSS-SEIDEL METHOD: EXAMPLE Cek pkh mtriks y digolly domit A Ber. Sehrusy koverge deg Guss-Siedel Method

20 GAUSS-SEIDEL METHOD: EXAMPLE Tulis ulg Asumsi ili wl

21 GAUSS-SEIDEL METHOD: EXAMPLE The bsolute reltive pproimte error % % % Glt bsolut terbesr di khir itersi pertm dlh %

22 GAUSS-SEIDEL METHOD: EXAMPLE Setelh itersi # Msukk ili pd persm Setelh itersi #

23 GAUSS-SEIDEL METHOD: EXAMPLE Glt bsolut dri Itersi # % % %.88 Glt bsolut mksimum 4.6% Lebih besr dri itersi #. Is this problem?

24 GAUSS-SEIDEL METHOD: EXAMPLE Ulgi itersi, didptk Itertio Hsil khir Medekti solusi sejti 4

25 LATIHAN Sistem persm liier With iitil guess of

26 GAUSS-SEIDEL METHOD The Guss-Seidel Method c still be used The coefficiet mtri is ot digolly domit A But this is the sme set of equtios used i emple #, which did coverge. A If system of lier equtios is ot digolly domit, check to see if rerrgig the equtios c form digolly domit mtri.

27 GAUSS-SEIDEL METHOD Not every system of equtios c be rerrged to hve digolly domit coefficiet mtri. Observe the set of equtios Which equtio(s) prevets this set of equtio from hvig digolly domit coefficiet mtri?

28 GAUSS-SEIDEL METHOD Summry -Advtges of the Guss-Seidel Method -Algorithm for the Guss-Seidel Method -Pitflls of the Guss-Seidel Method

29 GAUSS-SEIDEL METHOD Questios?

30 METODE PENYELESAIAN Metode grfik Elimisi Guss Metode Guss Jourd Metode Guss Seidel LU decompositio

31 LU DECOMPOSITION A=LU A=b LU=b Defie U=y Ly=b Solve y by forwrd substitutio U=y Solve by bckwrd substitutio

32 LU DECOMPOSITION BY GAUSSIAN ELIMINATION ) ( ) ( ) ( () () () () () () () () (),4,,,,,,,,, m m m m m m m m m m A Compct storge: The digol etries of L mtri re ll s, they do t eed to be stored. LU is stored i sigle mtri. There re ifiitely my differet wys to decompose A. Most populr oe: U=Gussi elimited mtri L=Multipliers used for elimitio

33 NEXT: SOLUSI PERSAMAAN NON LINIER Persm mtemtis yg sulit diselesik deg tg litis, sehigg diperluk peyelesi pedekt umerik Metode Numerik: Tekik meyelesik mslh mtemtik deg pegopersi hitug, umumy meckup sejumlh besr klkulsi ritmetik yg sgt byk d mejeuhk Diselesik deg lgoritm (sergki perith utuk meyelesik mslh), sehigg diperluk btu komputer utuk melksky

34 SUMBER GALAT / ERROR Keslh pemodel cotoh: peggu hukum Newto sumsi bed dlh prtikel Keslh bw cotoh: kekeliru dlm meyli dt slh membc skl Ketidktept dt Keslh pemotog / peyederh persm(tructio error) Keslh pembult (roud-off error)

35 SOLUSI PERSAMAAN NON LINEAR )Metode Akolde (brcketig method) / Closed method Metode Bgi du (Bisectio Method) Metode Regul Flsi (Flse Positio Method) Metode Grfik Kerugi: reltif lmbt koverge Keutug: sellu koverge

36 SOLUSI PERSAMAAN NON ) Metode Terbuk LINEAR Cotoh: Itersi Titik-Tetp (Fi Poit Itertio) Metode Newto-Rphso Metode Sect Keutug: cept koverge Kerugi: tidk sellu koverge