Solusi Sistem Persamaan Linear

Transkripsi

1 Sos Sstem Persm Ler Sstem persm er: h persm deg h kow j d dketh, j,,, j?

2 So: z 6 z z () () () persm d kow Jw: z 6.5 z.5 z () () () ems : pers. ().5 pers. () pers. ().5 pers. () z 6.5 z 8z 8 () () () ems : pers. () 5 pers. () z z.5 6 z sstts mdr: pers. () mecr z pers. () mecr pers. () mecr

3 Dm etk mtrks: So: Jw: 6 z 6 z z 8.5 z 6.5 z z

4 Ems Gss Metode Ems Gss mecr sos seh sstem persm er deg cr sepert dtjkk pd cotoh seem : k,...,),...,;j k ;,..., (k, ) (k k (k-) kk (k-) k ) (k (k) (k-) kj (k-) kk (k-) k (k-) (k) () () (-) () () () (-) () () () () () () () () () hm erkt (m) (m),, pd gkh ke m

5 ),..., (j (-j-) -j,-j -j k k (-j-) -j,k (-j-) -j j (-) (-) (-) () () () (-) () () () () () () () () () Sstts mdr:

6 A X B t AX B Jd, metode Ems Gss terdr dr d thp:. trgs: megh mtrks A mejd mtrks segtg (mtrks B deg egt jg erh). sstts mdr: meghtg megkt rt terk, dr g terkhr ( ) smp g pertm ( )

7 LU Decomposto A X B t AX B Pd metode LU Decomposto, mtrks A dts g seg perk mtrks L d U (mtrks A dr mejd mtrks L d U). Mtrks L d U merpk mtrks segtg. Mtrks B tdk erh, kre mtrks A tdk erh, mek h dts g. U A X B X L B

8 Lgkh:. Cr mtrks L d U sehgg A LU. Mtrks B tetp. A L U L U X B. Defsk seh mtrks koom r, ms Y, t Y UX, sehgg LY B. L htg deg sstts mj (m dr smp ). U Y X Y L B. Htg deg sstts mdr (m dr smp ). U X Y Jes hw metode LU Decomposto pd prsp sm deg metode Ems Gss: mtrks U merpk hs trgs mtrks A, g jg megktk B erh mejd Y.

9 Mecr mtrks L d U: Dperoeh:,...,) (j,...,) (,...,) (j,...,) (,...,) (j,...,) ( j j j j j j j j j j j j j j j j j j

10 Jd, eeme mtrks L d U dcr mert: j j j k k k k kj kj (,...,) (j,...,) ( j,...,; j,..., ) (j,...,;,..., -) secr ergt:. mtrks L koom, mtrks U rs. mtrks L koom, mtrks U rs.. mtrks L koom (-), mtrks U rs (-) 5. mtrks L koom

11 Sstts mj tk meghtg :,...,) ( j j Sstts mdr tk meghtg : ),..., ( j j,j -

12 Kem ke so Jw: 6 B, A A LU.5 U, L B LY UX, Y Y UX Y X B AX, deg.

13 Kss Beerp Sstem Persm Ler Pd kss g eh mm s sj terdpt eerp sstem persm er deg B g er, m memk A g sm. Dm etk mtrks sstem sepert dtsk seg: A X B t AX B Keterg: A mtrks, X d B mtrks m, deg m jmh sstem persm er, jmh persm / kow dm tp sstem persm terset Tp koom mtrks X merpk sos tk koom g sm pd mtrks B. Lgkh d rms pd metode Ems Gss d LU Decomposto erk sm tk kss. H sj, d s mtrks X d B terdr dr eerp koom, k h st.

14 Cotoh d sstem persm er g memk A sm tp B ered.

15 Metode Ems Gss:,...,m) k,...,;r j,...,; k ;,..., (k,...,m),...,;r (,j, ) (k kr (k-) kk (k-) k ) (k r (k) r (k-) kj (k-) kk (k-) k (k-) (k) r () r () r (m) r (m),, pd gkh ke m rms trgs: rms sstts mdr:,...,m) ;r,..., (j,...,m) (r (-j-) -j,-j -j k kr (-j-) -j,k (-j-) -j,r j,r (-) (-) r r

16 Metode LU Decomposto: rms sstts mj tk meghtg (k Y mtrks m): r r (r,...,m) r r j jr (,...,;r,...,m) rms sstts mdr tk meghtg : r -,r r,r j,j jr (r,...,m) (,..., ;r,...,m)

17 Perhtk metode LU Decomposto, ggp mtrks L d U teh dperoeh. Jk kemd terdpt g sstem persm er deg A sm d B ered, mk mtrks L d U g teh dperoeh t s gsg dpk tk sstem persm g r terset. K perhtk metode Ems Gss, ggp trgs mtrks A sdh dkerjk. Jk kemd terdpt g sstem persm er deg A sm d B ered, mk hs trgs mtrks A g sdh dperoeh tdk dpt dpk tk sstem persm g r. Utk sstem g r terset proses trgs mtrks A hrs dkk g dr w. H dsek, mtrks B hrs erh megkt proses trgs mtrks A, semetr proses pegr mtrks A mejd mtrks L d U tdk metk mtrks B.

18 Ctt: Dm rms-rms k pd metode Ems Gss mp LU Decomposto terdpt pemg oeh eeme dgo mtrks t, oeh eeme dgo mtrks A pd metode Ems Gss, d eeme dgo mtrks L pd metode LU Decomposto. Jk secr keet eeme dgo t o, mk k tm error. Kre t, pd setp gkh dm proses trgs mtrks A (metode Ems Gss) t pecr mtrks L d U (metode LU Decomposto) per dkk pemerks, pkh eeme mtrks A t L g ersgkt sm deg o. Jk er o, mk rs ers eeme dgo o t hrs dtkr deg sh st rs seteh, sehgg eeme dgo mejd k o. Perh rs pd mtrks A (metode Ems Gss) hrs dsert perh rs g sm pd mtrks B. Perh rs pd mtrks L (metode LU Decomposto) hrs dsert perh rs g sm pd mtrks A d B.

19 So: Jw: rs dtkr deg rs

20 5 A L.5.5 U L L B 5 A B U.75.5 L / U.75.5 L rs rs