Simultaneous Linear Equations

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Yulia Sudjarwadi
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1 Simultneous Liner Equtions Topic: Guss-Seidel Method 0//

2 Guss-Seidel Method Prosedur dsr: Adlh metode ITERASI - Menyelesikn tip persmn linier secr ljbr untuk i - Membut nili sumsi solusi - Selesikn untuk tip i dn ulngi - Gunkn perkirn keslhn reltif tip khir itersi untuk mengecek pkh error sudh mencpi ngk tolernsi.

3 Guss-Seidel Method Kenp? Untuk mengtsi round-off error (keslhn pembultn). Metode eliminsi seperti Gussin Elimintion nd LU Decomposition(*) rwn terhdp keslhn pembultn.

4 Guss-Seidel Method Sistem persmn linier Algorithm n n b n n b n n Kit mengubh sistem n nn n b n persmn [A]{X}{B} untuk menyelesikn dengn persmn pertm, menyelesikn dengn persmn kedu, dn seterusny.

5 Guss-Seidel Method Algorithm Generl Form of ech eqution c n j j, n j j j n n c j j c j n j j j, n n n j j n nn n n j j j nj n n c

6 Menjdi: Untuk sistem persmn b b b Now we cn strt the solution process by choosing guesses for the s. A simple wy to obtin initil guesses is to ssume tht they re zero. These zeros cn be substituted into eqution to clculte new b /.

8 Bts khir itersi New is substituted to clculte nd. The procedure is repeted until the convergence criterion is stisfied: lm i ε 00 pε i bru i bru i diperkenn kn ε ε t pproimtion error, sering digunkn, seringkli disebut sebgi glt bsolut. True error, kurng berrti. digunkn Reltive error, dlm prosentse

9 Guss-Seidel Method: Emple Dikethui sistem persmn Initil Guess: sumsi nili wl, 5

10 Guss-Seidel Method: Emple Tulis ulng untuk pliksi Guss-Seidel

11 Guss-Seidel Method: Emple Msukkn nili perkirn wl untuk selesikn i () (5) (.670) ( 5) Initil Guess (.670) ( 7.850) 55.6

12 Guss-Seidel Method: Emple Finding the bsolute reltive pproimte error ε new old i i i new i ε ε % % At the end of the first itertion The mimum bsolute reltive pproimte error is 5.47% ε % 55.6

13 Guss-Seidel Method: Emple Itertion # Using.670 the vlues of i re found: ( 7.850) from itertion # (.056) (.056) ( 54.88)

14 Guss-Seidel Method: Emple Hitung the bsolute reltive pproimte error %.056 ( 7.850) ( 55.6) % 80.54% Akhir itersi kedu Glt bsolut terbesr %

15 Guss-Seidel Method: Emple Tersukn itersi, kit dptkn nili berikut Itertion % % % ! Lho, kok? Error ny nggk berkurng?

16 Guss-Seidel Method: Pitfll Slhny dimn? Contoh tdi mengilustrsikn kemungkinn keslhn pd Guss- Siedel method: tidk semu sistem persmn kn konvergen. Is there fi? One clss of system of equtions lwys converges: One with digonlly dominnt coefficient mtri. Digonlly dominnt: [A] in [A] [X] [C] is digonlly dominnt if: ii n j j i ij Untuk semu i ; DAN ii ij n j j i Untuk miniml sebuh i

17 Guss-Seidel Method: Pitfll Digonlly dominnt: Koefisien pd digonl hrus sm tu lebih besr dri jumlh semu koefisien pd bris itu, dn miniml stu bris hrus memiliki digonl yng lebih besr dri jumlh koefisien pd bris itu. Mnkh mtriks yng digonlly dominnt? [ A ] [B]

18 Guss-Seidel Method: Emple Sistem persmn linier Dengn sumsi nili wl 0 Mtriks Koefisien ny dlh 5 [ A] 5 7 Akn konvergen kh?

19 Guss-Seidel Method: Emple Cek pkh mtriks ny digonlly dominnt [ A] Benr. Sehrusny konvergen dengn Guss-Siedel Method

20 Guss-Seidel Method: Emple Tulis ulng Asumsi nili wl ( ) ( ) ( ) ( ) ( ) ( )

21 Guss-Seidel Method: Emple The bsolute reltive pproimte error % % % Glt bsolut terbesr di khir itersi pertm dlh 00%

22 Guss-Seidel Method: Emple Setelh itersi # Msukkn nili pd persmn Setelh itersi # ( ) + 5(.09) 8 ( ) (.09) 76 5 ( ) 7( 4.900)

23 Guss-Seidel Method: Emple Glt bsolut dri Itersi # % % % Glt bsolut mksimum 40.6% Lebih besr dri itersi #. Is this problem?

24 Guss-Seidel Method: Emple Ulngi itersi, didptkn Itertion ε ε ε Hsil khir.000 Mendekti solusi sejti

25 Ltihn Sistem persmn linier With n initil guess of 0

26 Guss-Seidel Method The Guss-Seidel Method cn still be used The coefficient mtri is not [ A] digonlly dominnt But this is the sme set of equtions used in emple #, which did converge. [ A] If system of liner equtions is not digonlly dominnt, check to see if rerrnging the equtions cn form digonlly dominnt mtri.

27 Guss-Seidel Method Not every system of equtions cn be rerrnged to hve digonlly dominnt coefficient mtri. Observe the set of equtions Which eqution(s) prevents this set of eqution from hving digonlly dominnt coefficient mtri?

28 Guss-Seidel Method Summry -Advntges of the Guss-Seidel Method -Algorithm for the Guss-Seidel Method -Pitflls of the Guss-Seidel Method

29 Guss-Seidel Method Questions?

30 Metode Penyelesin Metode grfik Eliminsi Guss Metode Guss Jourdn Metode Guss Seidel LU decomposition

31 LU Decomposition ALU Ab LUb Define Uy Lyb Solve y by forwrd substitution Uy Solve by bckwrd substitution

32 LU Decomposition by Gussin elimintion A There re infinitely mny different wys to decompose A. Most populr one: UGussin eliminted mtri LMultipliers used for elimintion m m M m n m () () () () 0 0 L 0 L n, ( ) ( ) ( ) 0 L 0 L n, () () m, L 0 0 L n, n, m m M n, n, m O m n, n, O L m n,4 M L M M 0 0 M 0 0 M 0 0 O ( n ) nn Compct storge: The digonl entries of L mtri re ll s, they don t need to be stored. LU is stored in single mtri. 0 M ( n ) nn ( n ) nn

33 NEXT: Solusi persmn Non Linier Persmn mtemtis yng sulit diselesikn dengn tngn nlitis, sehingg diperlukn penyelesin pendektn numerik Metode Numerik: Teknik menyelesikn mslh mtemtik dengn pengopersin hitungn, umumny menckup sejumlh besr klkulsi ritmetik yng sngt bnyk dn menjenuhkn Diselesikn dengn lgoritm (serngkin perinth untuk menyelesikn mslh), sehingg diperlukn bntun komputer untuk melksnknny

34 Sumber Glt / Error Keslhn pemodeln contoh: penggunn hukum Newton Keslhn bwn sumsi bend dlh prtikel contoh: kekelirun dlm menylin dt Ketidkteptn dt slh membc skl Keslhn pemotongn / penyederhnn persmn(trunction error) Keslhn pembultn (round-off error)

35 Solusi Persmn Non Liner )Metode Akolde (brcketing method) / Closed method Metode Bgi du (Bisection Method) Metode Regul Flsi (Flse Position Method) Metode Grfik Kerugin: reltif lmbt konvergen Keuntungn: sellu konvergen

36 Solusi Persmn Non Liner ) Metode Terbuk Contoh: Itersi Titik-Tetp (Fi Point Itertion) Metode Newton-Rphson Metode Secnt Keuntungn: cept konvergen Kerugin: tidk sellu konvergen

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