MetodeLelaranUntukMenyelesaikanSPL

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1 MetodeLelrUtukMeyelesikSPL Metode elimisi Guss melitk yk glt pemult. Glt pemult yg terjdi pd elimisi Guss dpt meyek solusiyg diperoleh juh drisolusiseery. Ggs metod lelr pd pecri kr persm irljr dptjugditerpkutukmeyelesikspl. Deg metode lelr, glt pemult dpt diperkecil, kre kitdptmeerusklelrsmpisolusiysetelitimugki, sesuidegtsgltyg kitperolehk. Degktli, esrgltdptdikedliksmpitsyg is diterim Rildi Muir - Topik Khusus Iformtik I 7

2 Jik metode elimisi Guss d vrisi-vrisiy sert metodedekomposisiludimkmetodelgsug(direct) -kre solusi SPL diperoleh tp lelr- mk metode lelr dimk metode tidk lgsug (idirect) tumetodeitertif. Tiju kemli sistem persm ljr = = 2 : : = Degsyrt kk 0, k =, 2,...,, mkpersm lelry dpt ditulis segi Rildi Muir - Topik Khusus Iformtik I 8

3 (k+) = k ( k ) 2 (k+) = M (k+) = ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) deg k = 0,, 2, Rildi Muir - Topik Khusus Iformtik I 9

4 Lelr dimuli deg memerik tek wl utuk, 0 = 2 M (0) (0) (0) Segi kodisi erheti lelry, dpt diguk pedekt glt reltif i ( k + ) ( k ) i ( k + ) i < ε utuk semu i =, 2, 3,., Syrt cukup gr lelry koverge dlh sistem domi secr digol: ii > j=, j i ij, i =, 2, 3,, Rildi Muir - Topik Khusus Iformtik I 20

5 Segi cotoh, SPL erikut = = = 5 domi secr digol, kre 3 > + - > > kre itu lelry psti koverge. Ad du metode lelr yg k kit hs di sii:. Metode lelr Jcoi 2. Metode lelr Guss-Seidel Rildi Muir - Topik Khusus Iformtik I 2

6 Metode Lelr Jcoi Persm lelry dlh seperti yg ditulis di ts. Mislkdieriktekwl (0) : (0) = ( (0), 2 (0),..., (0) ) T Prosedur lelr utuk lelr pertm, kedu, d seterusy dlh segi erikut: Rildi Muir - Topik Khusus Iformtik I 22

7 Lelr pertm: () = ( 0) ( 0) ( 0) () M = ( 0) ( 0) ( 0) () = ( 0) ( 0) ( 0) Rildi Muir - Topik Khusus Iformtik I 23

8 Lelr kedu: (2) = ( ) ( ) ( ) (2) M = ( ) ( ) ( ) (2) = ( ) ( ) ( ) Rumus umum : i ( k + ) = i ii ij j=, j i j ( k ), k = 0,,2,... Rildi Muir - Topik Khusus Iformtik I 2

9 Metode Lelr Guss-Seidel Kecept koverge pd lelr Jcoi dpt diperceptilsetiphrg i yg rudihsilk seger dipki pd persm erikuty utuk meetukhrg i+ yg liy. Metode lelry dimk lelr Guss-Seidel Rildi Muir - Topik Khusus Iformtik I 25

10 Lelr pertm: () = 2 2 ( 0) ( 0) ( 0) () = 2 ( ) ( 0) ( 0) () = 3 3 ( ) ( ) ( 0) () = ( ) ( ) ( ) Rildi Muir - Topik Khusus Iformtik I 26

11 Lelr kedu: (2) = 2 (2) = ( ) ( ) ( ) 3 3 ( 2) ( ) ( ) (2) = 3 3 ( 2) ( 2) ( ) (2) = ( 2) ( 2) ( 2) Rumus umum: i ( k + ) = i j= ij j ( k + ) ( k ) j= i+ ij j, k = 0,,2,... ii Rildi Muir - Topik Khusus Iformtik I 27

12 Cotoh: TetuksolusiSPL -y+ z= 7-8y+ z= y+ 5z= 5 degiliwlp 0 = ( 0, y 0, z 0 ) = (, 2, 2). (Solusisejtiy dlh(2,, 3) ) Peyelesi: () Metode lelr Jcoi Persm lelry: r+ = y r+ = z r+ = 7 + yr zr 2+ r zr r yr 5 Rildi Muir - Topik Khusus Iformtik I 28

13 Lelry: = y = z = ( ) 8 2( ) =.75 = = = y 2 = z 2 = ( 3.375) ( ) =.8375 = = = y 9 = z 9 = Rildi Muir - Topik Khusus Iformtik I 29

14 () Metode lelr Guss-Seidel Persm lelry, r+ = y r+ = z r+ = 7 + yr zr 2 + r zr r yr 5 Lelry, = y = z = ( ) 8 ( ) =.75 = 3.75 = Rildi Muir - Topik Khusus Iformtik I 30

15 2 = y 2 = z 2 = ( ) =.95 = = = y 0 = z 0 = Jdi, solusi SPL dlh = , y = , z = Rildi Muir - Topik Khusus Iformtik I 3

16 Cotoh Sol Terp Dlm hl ii, semu rus i yg memsuki simpul diggp ertd positif. Sedgk hukum Ohm meytk hw rus i yg mellui sutu th dlh : i ij = V V i R ij j yg dlm hl ii V dlh tegg d R dlh th. i i 2 R ij V i V j i 3 i ij rh rus () () Rildi Muir - Topik Khusus Iformtik I 32

17 Dierik seuh rgki listrik deg 6 uh th seperti pd Gmr di wh ii. Ad dimit meghitug rus pd msig-msig rgki. 3 R 32 R 2 2 i 32 i 2 R 3 i 3 i 52 R 52 i 5 i 65 R 5 5 R 65 6 Rildi Muir - Topik Khusus Iformtik I 33

18 Peyelesi: Arh rus dimislk seperti dits. Deg hukum Kirchoff diperoleh persm-persm erikut: i 2 + i 52 + i 32 = 0 i 65 - i 52 - i 5 = 0 i 3 - i 32 = 0 i 5 - i 3 = 0 Dri hukumohm didpt: i 32 R 32 -V 3 + V 2 = 0 i 3 R 3 -V + V 3 = 0 i 65 R 65 + V 5 = 0 i 2 R 2 + V 2 = 0 i 5 R 5 -V 5 + V = 0 i 52 R 52 -V 5 + V 2 = 0 Deg meyusu kesepuluh persm dits didptk SPL s: Rildi Muir - Topik Khusus Iformtik I 3

19 i 2 i 52 i 32 i 65 i 5 i 3 V 2 V 3 V V i i i i R i R i 3 = R V 2 V 6 R V 3 V R V 0 0 R V 5 0 Tetuk il dikethui i 2, i 52, i 32, i 65, i 5, i 3, V 2, V 3, V, V 5 R 2 = 5 ohm, R 52 = 0 ohm, R 32 = 0 ohm R 65 = 20 ohm, R 5 = 5 ohm, R = 5 ohm. V = 200 volt, V 6 = 0 volt. Rildi Muir - Topik Khusus Iformtik I 35

20 Persol ii diselesik deg metode elimisi Guss. Mtriks wl seelum proses elimisi Guss dlh: Rildi Muir - Topik Khusus Iformtik I 36

21 Mtriks khir setelh elimisi dlh: Deg tekik peyulih mudur diperoleh solusiy segi erikut: i2 =. mpere, i52 = -. mpere i32 = mpere, i65 = mpere i5 = mpere, i3 = mpere V2 = volt, V3 = volt V = volt, V5 = volt Rildi Muir - Topik Khusus Iformtik I 37

22 Rildi Muir - Topik Khusus Iformtik I 38