SISTEM PERSAMAAN LINEAR

http://istirto.stff.ugm..id SISTEM PERSAMAAN LINEAR Systems of Lier Algebri Equtios

Sistem Persm Lier http://istirto.stff.ugm..id Au Chpr, S.C., Cle R.P., 99, Numeril Methods for Egieers, d Ed., MGrw-Hill Book Co., New York. Chpter 7, 8, d 9, hlm. -9.

Sistem Persm Lier http://istirto.stff.ugm..id Sergki persm lier:............ Sejumlh persm lier ii hrus diselesik ser simult utuk medptk,,, yg memeuhi setip persm tsb.

4 Metode Peyelesi http://istirto.stff.ugm..id Jml. pers. sedikit, «Jml. pers. byk,» Peyelesi Peyelesi lgsug Grfis Elimisi Guss Crmer Guss-Jord Elimisi Itertif Jobi Guss-Seidel Suessive Over Reltio

5 Metode Grfis http://istirto.stff.ugm..id

6 Metode Grfis http://istirto.stff.ugm..id hmpir sejjr berimpit sejjr ill-oditioed system sigulr system sigulr system

7 Metode Crmer http://istirto.stff.ugm..id Vribel tk dikethui, i, merupk perbdig du determi mtriks Peyebut : determi, D, mtriks koefisie sistem persm Pembilg : determi mtriks koefisie sistem persm seperti peyebut, mu koefisie kolom ke i digti deg koefisie i Cotoh persm lier

http://istirto.stff.ugm..id Metode Crmer 8 [ ] A A det D A D D D

9 Determi Mtriks http://istirto.stff.ugm..id Mtriks bujur sgkr: Meri determi mtriks Hitug mul MSEel, deg fugsi MDETERM() Cotoh hitug determi mtriks d [ A] A [ B] B

http://istirto.stff.ugm..id Determi Mtriks det D A ( ) ( ) ( ) det D B

Metode Crmer http://istirto.stff.ugm..id Cotoh: persm lier A X C... 7... 7.85 9. 7.4... 7... 7.85 9. 7.4 deta ( ) [( 7)( ) (.)(.) ] (.) [(.)( ) (.)(.) ] (.) [(.)(.) ( 7)(.) ].5

Metode Crmer http://istirto.stff.ugm..id 7.85. 7.4.. [ A ] 9. 7. [ A ]. 9.. [ ]. 7.85 7.4. A... 7. 7.85 9. 7.4 deta A 6.59 deta A 55. 88 det A 47. 47 deta 6.59 deta 55.88 deta 47.47. 5 7 deta.5 deta.5 deta.5 A

http://istirto.stff.ugm..id Metode Elimisi Cotoh: persm lier [ ] [ ]

http://istirto.stff.ugm..id Elimisi Guss 4 Strtegi Forwrd elimitio Bk substitutio Cotoh persm lier () () ()

5 Elimisi Guss http://istirto.stff.ugm..id Forwrd elimitio # Hilgk dri pers. kedu d ketig deg opersi perkli koefisie d pegurg deg pers. pertm. pivot oeffiiet pivot equtio ʹ ʹ ʹ ʹ ʹ ʹ () (') (')

6 Elimisi Guss http://istirto.stff.ugm..id Forwrd elimitio # Hilgk dri pers. ketig deg opersi perkli koefisie d pegurg deg pers. kedu. pivot oeffiiet ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ pivot equtio ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ () (') ('')

7 Elimisi Guss http://istirto.stff.ugm..id Bk substitutio Hitug dri pers. (''), hitug dri pers. ( ), d dri pers. () ʹ ʹ ʹ ʹ ʹ ʹ ʹ

http://istirto.stff.ugm..id Elimisi Guss 8 Forwrd elimitio............ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ,...,,, i i ii i j j i ij i i i Bk substitutio

http://istirto.stff.ugm..id Elimisi Guss 9 7.4.. () 9.. 7. () 7.85.. () Cotoh: persm lier

Elimisi Guss http://istirto.stff.ugm..id Forwrd elimitio Elimisi dri Pers. d, Pers. sebgi pivot () ( ʹ ) (ʹ ʹ ) Elimisi dri Pers., Pers. sebgi pivot ().. 7.85 ( ʹ ) (ʹ ʹ ). 7..9 7....9.9. 7.85 7.65 9.567 9.567 7.84

Elimisi Guss http://istirto.stff.ugm..id Bk substitutio Meghitug dri Pers. '' 7.84. 7 Substitusi ke Pers. ' utuk meghitug 9.567.9 7. ( 7).5 Substitusi d ke Pers. utuk meghitug 7.85. ( 7).(.5)

Metode Elimisi http://istirto.stff.ugm..id Strtegi Elimisi vribel tk dikethui, i, deg peggbug du persm. Hsil elimisi dlh stu persm yg dpt diselesik utuk medptk stu vribel i.

Kelemh Metode Elimisi http://istirto.stff.ugm..id Pembgi deg ol Pivot oeffiiet sm deg ol tupu sgt keil. Pembgi deg ol dpt terjdi selm proses elimisi tupu substitusi. Roud-off errors Selm proses elimisi mupu substitusi, setip lgkh hitug bergtug pd lgkh hitug sebelumy d setip kli terjdi keslh; keslh dpt berkumulsi, terutm pbil jumlh persm sgt byk. Ill-oditioed systems Ill-oditio dlh situsi dim perubh keil pd stu tu beberp koefisie berkibt perubh yg besr pd hsil hitug.

4 Perbik http://istirto.stff.ugm..id Pemilih pivot (pivotig) Urut persm dipilih sedemiki higg yg mejdi pivot equtio dlh persm yg memberik pivot oeffiiet terbesr.

5 Metode Peyelesi http://istirto.stff.ugm..id Mtriks Iversi Guss-Jord Metode Itertif Jobi Guss-Seidel

6 Metode Guss-Jord http://istirto.stff.ugm..id Mirip deg metode elimisi Guss, tetpi tidk diperluk bk substitutio. Cotoh persm lier... 7... 7.85 9. 7.4

http://istirto.stff.ugm..id Metode Guss-Jord 7 7.4 9. 7.85... 7... 7.4 9. 7.85... 7... 7.4 9..667... 7..667. 7.65 9.567.667..9.9 7..667.

http://istirto.stff.ugm..id 7.84.79.56..49.68 Metode Guss-Jord 8 7.65 9.567.667..9.9 7..667. 7.65 9.567/ 7..667..9.9/ 7. 7./ 7. / 7..667. 7.65.79.667..9.49.667.

http://istirto.stff.ugm..id 7.79.56.49.68 7.84/..79.56./. /. /..49.68 Metode Guss-Jord 9 7.5 7.5

Guss-Jord vs Elimisi Guss http://istirto.stff.ugm..id Metode Guss-Jord Jumlh opersi lebih byk (5%) Memiliki kelemh yg sm deg elimisi Guss Pembgi deg ol Roud-off error

Mtriks Iversi http://istirto.stff.ugm..id [ A] { X} { C} { X} [ A] { C}

http://istirto.stff.ugm..id Mtriks Iversi 7.4.. () 9.. 7. () 7.85.. () Cotoh: persm lier

http://istirto.stff.ugm..id Mtriks Iversi [ ]... 7... A [ ].... 7..667. A [ ].999....9.9 7..667. A

http://istirto.stff.ugm..id Mtriks Iversi 4 [ ].999.4.47...9.47.667. A [ ].7.9.4.47.47.8..47.68 A

http://istirto.stff.ugm..id Mtriks Iversi 5 [ ].999.7..4.47.47.8.47.68 A [ ].999.7..4.4.5.68.49.5 A [ ] A

http://istirto.stff.ugm..id Mtriks Iversi 6 { } [ ] { } 7..488.4 7.4 9. 7.85.999.7..4.4.5.68.49.5 C A X

http://istirto.stff.ugm..id Metode Itertif: Jobi 7 ili wl, bisy i itersi diterusk smpi koverge i i, i

8 Metode Itertif: Jobi http://istirto.stff.ugm..id Cotoh: persm lier () () ()... 7... 7.85 9. 7.4

http://istirto.stff.ugm..id Metode Itertif: Guss-Seidel 9 itersi diterusk smpi koverge i i, i

4 Metode Itertif: Guss-Seidel http://istirto.stff.ugm..id Cotoh: persm lier () () ()... 7... 7.85 9. 7.4

http://istirto.stff.ugm..id Jobi vs Guss-Seidel 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Jobi Guss-Seidel

4 Suessive Over-reltio Method http://istirto.stff.ugm..id Dlm setip itersi, ili vribel terbru (yg bru sj dihitug),, tidk lgsug dipki pd itersi seljuty Pd itersi seljuty, ili tsb dimodifiksi deg memsukk pegruh ili vribel lm (pd itersi sebelumy), ew λ i ( λ) i fktor relkssi λ dimksudk utuk memperept kovergesi hitug (itersi) uder-reltio: < λ < over-reltio: < λ <

http://istirto.stff.ugm..id Suessive Over-reltio Method 4 ( ) [ ] ( ) [ ] ( ) [ ] λ λ λ λ λ λ

44 Suessive Over-reltio Method http://istirto.stff.ugm..id Cotoh: persm lier () () ()... 7... 7.85 9. 7.4

45 http://istirto.stff.ugm..id