SISTEM PERSAMAAN LINEAR Systems of Lier Algebri Equtios
Sistem Persm Lier 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 Sergki persm lier:............ Sejumlh persm lier ii hrus diselesik ser simult utuk medptk,,, yg memeuhi setip persm tsb.
Metod Peyelesi 4 Jml. pers. sedikit, «Jml. pers. byk,» Peyelesi Peyelesi lgsug Grfis Elimisi Guss Crmer Guss-Jord Elimisi Itertif Jobi Guss-Seidel Suessive Over Reltio
5 Metod Grfis = =
6 Metod Grfis hmpir sejjr berimpit sejjr ill-oditioed system sigulr system sigulr system
7 Metod Crmer Vribel tk dikethui, i, merupk perbdig du determi mtri Peyebut : determi, D, mtri koefisie sistem persm Pembilg : determi mtri koefisie sistem persm seperti peyebut, mu koefisie kolom ke i digti deg koefisie i Cotoh persm lier
Metod Crmer 8 A A det D A D D D
9 Determi Mtri Mtri bujur sgkr: Meri determi mtri Hitug mul MSEel, deg fugsi =MDETERM() Cotoh hitug determi mtri d A A B B
Determi Mtri det D A det D B
Metod Crmer Cotoh: persm lier A X C... 7... 7.85 9. 7.4... 7... 7.85 9. 7.4 deta 7......... 7..5
Metod Crmer 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
Metod Elimisi Cotoh: persm lier
Elimisi Guss 4 Strtegi Forwrd elimitio Bk substitutio Cotoh persm lier () () ()
5 Elimisi Guss Forwrd elimitio # Hilgk dri pers. kedu d ketig deg opersi perkli koefisie d pegurg deg pers. pertm. pivot oeffiiet pivot equtio () (') (')
6 Elimisi Guss Forwrd elimitio # Hilgk dri pers. ketig deg opersi perkli koefisie d pegurg deg pers. kedu. pivot oeffiiet pivot equtio () (') ('')
7 Elimisi Guss Bk substitutio Hitug dri pers. (''), hitug dri pers. ( ), d dri pers. ()
Elimisi Guss 8 Forwrd elimitio............,...,,, i i ii i j j i ij i i i Bk substitutio
Elimisi Guss 9 7.4.. () 9.. 7. () 7.85.. () Cotoh: persm lier
Elimisi Guss Forwrd elimitio Elimisi dri Pers. d, Pers. sebgi pivot () ( ) ( ) Elimisi dri Pers., Pers. sebgi pivot () ( ) ( ). 7..9. 7.....9.9. 7.85 7.85 7.65 9.567 9.567 7.84
Elimisi Guss Bkwrd 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
Metod Elimisi Strtegi Elimisi vribel tk dikethui, i, deg peggbug du persm. Hsil elimisi dlh stu persm yg dpt diselesik utuk medptk stu vribel i.
Kelemh Metod Elimisi 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 terkumulsi, terutm pbil jumlh persm sgt besr. Ill-oditioed systems Ill-oditio dlh situsi dim perubh keil pd stu tu beberp koefisie berkibt perubh yg besr pd hsil hitug.
Perbik 4 Pemilih pivot (pivotig) Urut persm dipilih sedemiki higg yg mejdi pivot equtio dlh persm yg memberik pivot oeffiiet terbesr.
Metod Peyelesi 5 Mtri Iverse Guss-Jord Metod Itertif Jobi Guss-Seidel
6 Metod Guss-Jord Mirip deg metod elimisi Guss, tetpi tidk diperluk bk substitutio. Cotoh persm lier... 7... 7.85 9. 7.4
Metod 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.
7.84.79.56..49.68 Metod 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.
7.79.56.49.68 7.84/..79.56./. /. /..49.68 Metod Guss-Jord 9 7.5 7.5
Guss-Jord vs Elimisi Guss Metod Guss-Jord Jumlh opersi lebih byk (5%) Memiliki kelemh yg sm deg elimisi Guss Pembgi deg ol Roud-off error
Iversi Mtri A X C X A C
Iversi Mtri 7.4.. () 9.. 7. () 7.85.. () Cotoh: persm lier
Iversi Mtri... 7... A.... 7..667. A.999....9.9 7..667. A
Iversi Mtri 4.999.4.47...9.47.667. A.7.9.4.47.47.8..47.68 A
Iversi Mtri 5.999.7..4.47.47.8.47.68 A.999.7..4.4.5.68.49.5 A A
Iversi Mtri 6 7..488.4 7.4 9. 7.85.999.7..4.4.5.68.49.5 C A X
Metod Itertif: Jobi 7 ili wl, bisy i = itersi diterusk smpi koverge i + i, i
8 Metod Itertif: Jobi Cotoh: persm lier () () ()... 7... 7.85 9. 7.4
Metod Itertif: Guss-Seidel 9 itersi diterusk smpi koverge i + i, i
4 Metod Itertif: Guss-Seidel Cotoh: persm lier () () ()... 7... 7.85 9. 7.4
Jobi vs Guss-Seidel 4 Jobi Guss-Seidel
4 Suessive Over-reltio Method 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 l i l i fktor relsi l dimksudk utuk memperept kovergesi hitug (itersi) uder-reltio: < l < over-reltio: < l <
Suessive Over-reltio Method 4 l l l l l l
44 Suessive Over-reltio Method Cotoh: persm lier () () ()... 7... 7.85 9. 7.4