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Penyelesaian Persamaan Linear Simultan
Metode Eliminasi Gauss Seidel
Ahmad Zainudin, S.ST, M.T
Workshop Metode Numerik
2014
Konsep Metode Eliminasi Gauss Seidel
Metode Eliminasi Gauss Seidel adalah metode
yang menggunakan proses iterasi hingga diperoleh
nilai-nilai yang berubah.
Bila diketahui persamaan linear simultan :
Konsep Metode Eliminasi Gauss Seidel
Hitung nilai xi untuk (i=1 s/d n)
Proses iterasi dihentikan bila selisih nilai xi (i=1 s/d
n) dengan nilai xi pada iterasi sebelumnya kurang
dari nilai yoleransi error yang ditentukan
Algoritma Metode Eliminasi Gauss Seidel
Program Metode Eliminasi Gauss Seidel
Tentukan Ordo dan Masukkan matrik A
Masukkan matrik B
Program Metode Eliminasi Gauss Seidel
Cetak Augmented Matrik [A|B]
Masukkan jumlah iterasi, toleransi error dan nilai awal
Program Metode Eliminasi Gauss Seidel
Cetak Header dan Tabel
Program Metode Eliminasi Gauss Seidel
Proses Iterasi dan Proses Gauss-Seidel
Program Metode Eliminasi Gauss Seidel
Cetak Hasil
Program Metode Eliminasi Gauss Seidel
Tampilan Program
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Metode Eliminasi Gauss Seidel

  • 1. Penyelesaian Persamaan Linear Simultan Metode Eliminasi Gauss Seidel Ahmad Zainudin, S.ST, M.T Workshop Metode Numerik 2014
  • 2. Konsep Metode Eliminasi Gauss Seidel Metode Eliminasi Gauss Seidel adalah metode yang menggunakan proses iterasi hingga diperoleh nilai-nilai yang berubah. Bila diketahui persamaan linear simultan :
  • 3. Konsep Metode Eliminasi Gauss Seidel Hitung nilai xi untuk (i=1 s/d n) Proses iterasi dihentikan bila selisih nilai xi (i=1 s/d n) dengan nilai xi pada iterasi sebelumnya kurang dari nilai yoleransi error yang ditentukan
  • 5. Program Metode Eliminasi Gauss Seidel Tentukan Ordo dan Masukkan matrik A Masukkan matrik B
  • 6. Program Metode Eliminasi Gauss Seidel Cetak Augmented Matrik [A|B] Masukkan jumlah iterasi, toleransi error dan nilai awal
  • 7. Program Metode Eliminasi Gauss Seidel Cetak Header dan Tabel
  • 8. Program Metode Eliminasi Gauss Seidel Proses Iterasi dan Proses Gauss-Seidel
  • 9. Program Metode Eliminasi Gauss Seidel Cetak Hasil
  • 10. Program Metode Eliminasi Gauss Seidel Tampilan Program
  • 11. 0 1 2 3 4 5 6 84 9 5 5 2 62 ! # $ $ # $ # % % # ' ' ' ' ( ) ( ) 6 1 2 3 4 5 6 * + ! # , $ # % % # - $ # ) ( ) ) ( ) ' ' 6 1 2 3 4 5 6 . + $ / 0 % , $ # % 1 % # % # $ # ) ( ) ) ( ) ' ' 6 1 2 3 4 5 6 + % 2 $ 2 , $ # % % # $ 2 $ / $ # ) 3 4 5 6 7 8 ( ) ) ( ) ' ' 6 1 2 3 4 5 6 9 + : # % # / , $ # % % # % # / % 2 $ # ) 3 4 5 6 7 8 ( ) ) ( ) ' ' 6 1 2 3 4 5 6 ; + # / # : $ , $ # % % # $ : # $ # ) 3 4 5 6 7 8 ( ) ) ( ) ' ' 6 1 2 3 4 5 6 = + / $ / # % , $ # % % # # % # / # $ # ) 3 4 5 6 7 8 ( ) ) ( ) ' ' 6 1 2 3 4 5 6 ? + % # $ 2 / $ 2 , $ # % % # / $ 2 / $ / $ # ) 3 4 5 6 7 8 ( ) ) ( ) ' ' 22 5 4 6 @ 4 A5 6 5 1 2 BC 2 3 5 4 B4 4 A6 A 6 2 3 + # % % $ $ # $ # ) D E6 4 6 6 A 1 2 3 4 5 6 @ 2? 5 94 F1 64 @G 2 3 G 24H 4 9 F 2 A I 4 AA 6 4 6 6 A 1 2 3 4 5 6 @ 2= B4 @ 4C 3 J 5 2 5 6 F 2 A 1 6 @ 4 A 4 A 6 C 2 3 J2 FC 2 A K 22 5 4 6 4 A + ) D ' ' ' ' A 6 4 6 4 L4+ M * N O 4 AM . N O
  • 13. 0 1 2 3 4 5 6 84 9 5 5 2 62 !26 5 1 2 #2 3 5 4 4 4 $% 6 $ 6 2 3 6 91 4 $ '4 3 6 4 ( 2) 4 $ * 6 + 4 3 6 444 ,- 9 4 ,( ! $ 2 . 4 / 4 $ * * 4 0 1 2444 ,- 9 4 ,( ! $ 2 . 43 1 4444 ,- 9 4 ,( ! $ 2 . 45 #2 3 , 4 1 6 . 4 $ 4 3 6 0 2 4 . 4 6 4 $( 4 , 4 $ 56 6 7 ( 4 , 4 $9 $ 1 9 .( ! $ 2 . 43 8 9 ( 4 , 4 $9 $ 1 9 .( ! $ 2 . 45 : ; 7 5 ( 4 , 4 $9 $ 1 9 .( ! $ 2 . 43 8 ( 4 , 4 $9 $ 1 9 .( ! $ 2 . 45 : = 6 0 2 3 !2 ,!2 5 6 5 1 2 0 2 3 5 4 4 4 $6 $ 6 2 3 2 ? @ 1 2A @ B @ 1 4C @ D ? 4 @ 1 2A @ E @ 1 4C @ F E 3 G H2 ( 9 4 1 4 + 4 ( ! $ 2 . 434 $5G 5! $ 2 . 43 2 2 3 9 . 4 $ ( 4 , 4 $6 7 (! .56 4 $ (! .5 / 5 24 $ * . 4 $( ! $ 2 . 45 2 2 3 9 . 4 $ ( 4 , 4 $9 (! .56 4 $ (! .5 G 52 3 4 0 4- 9 4 ,( ! $ 2 . 4) 4 $ * 4 0 4 1 6 , 4 5 6 . 4 $( 6 41 2 3 5 26 4; 7 (! .( 4 , 4 $56 4 $= (! .( 4 , 4 $5 G
  • 14. 0 1 2 3 4 5 6 84 9 5 5 2 62 2 ! 22 5 4 6 4 ! 2 ! # 4 !$2 ! # # 9 ! 4 % 4 ! $2 1 226 $6 ! 4 5 6 84 9 5 5' 34 ! 444 (5 2 ) 4 # 4 6 ) 2 3 6 % 9 1 * +9 # 2 $2 ! 1 2 ,4 1 3 6 % - . / 0 . 1 2 3 2 4- 5 4- 6 - . - . 7 / 0 1 2 3 2 41 5 41 1 4- - . 7 / 0 . / 1 . 41 5 41 6 / - . 7 / 0 . - 8 4- 5 4- . 7 / 41 - . 2 . - 8 96 : 2 3 2 (; - 2 4 !; 1 8 7 4 3 1 6 ! 4 ) 4 ( 4 ! 4 ! # 1 2 3 5 26 4 4 : 4 16 ) 9 4 1 2 ) ! 2 % 4+4 !8 ) ! 2 % 44=
  • 15. 0 1 2 3 4 5 6 84 9 5 5 2 62 2 3 4 5 44 4 46 3 4 4 4 5 444 1 4 4 6 2 1 4 9 6 4 4 5 2 2 3 4 41 6 1 6 444 1 4 4 4 6 1 9 445 6 5 69 4 3 6 4 6 1 2 1 9 4 4 446 3 4 4 4 5 2 3 2 3 4 5 2 4 3 44 6 4 3 4 4 4 5 445 2 9 4 1 6 1 6 444 3 4 1 43 4 1 4 4 4 5 4 3 6 1 6 1 6 1 2 1 4 4 4 4 4 4 4 16 6 1 9 4 4 5 441 6 1 6 ! 4 5 2 4 4 6 2 3 6 9 1 # $ % ' ( % ' ( % ' ( % ' ( % ) $ $ % *+ *, - . - . ) $ $ ' ( ' ( ' ( $ ' ( ) / ) ' ' / ) ) 0 0 0 1 0 0 0 1 2 2 2 2 6 5 1 2 2 3 5 4 4 4 6 6 2 3 4 3 6 2 3 4 5 44 4 6 4 1 4 5444 # 2 3 5 4 4 4 4 4 5 441 6 1 6 34 4 35 4 4 16 6 1 9 2 4 # ) ( $ / ( $ / ' ) ' ) 1 0 6 1 6 2 2 2 2
  • 16. 0 1 2 3 4 5 6 84 9 5 5 2 62 2 3 4 5 44 4 46 3 4 4 4 5 444 1 4 4 2 22 5 4 6 4 2 4 2 9 4 4 6 1 2 3 4 5 6 84 9 5 5 2 62 1 2 3 2 6 4 99 6 1 2 1 9 4 6 4 6 2 2 4 1 4 4 4 4 4 ! 9 5 6 2 9 4 4444 # $ # $ % % % ' ( ) % ' ( ) % * * * * + , + , -6 2 3 .2 4 5 6 2 3 6 1 9 4 9 1 9 1 .2 3 4 5 6 2 3 3 . 3 / 0 / / / 1 5 2 4 4 6 2 3 6 9 1 2 3 4 5 6 7 8 9 : 9 ; 4 : 4 ; / / / 1 1 = / ? = 1 = / ? = ? = ? = @ 1 ? / ? 1 / 1 = ? = = @ / ? @ @ @ 1 / @ @ A = / ? @ A ? ? / 1 = B ? 1 @ @ @ / ? @ @ @ ? A / / @ ? A @ / / / 1 B = @ @ @ @ 1 A @ @ @ @ / / / B / / / / ? ? @ @ @ @ @ @ @ @ @ @ @ / / / / 1 = = @ B A C/ = A @ @ @ @ @ @ @ @ @ @ @ @ ? C/ ? C/ ? D 46 1 2 2 3 4 1 9 3 44 E1 F @ @ @ @ @ 4 E F @ @ @ @ @
  • 17. 0 1 2 3 4 5 6 84 9 5 5 2 62 ! # $2 3 % 4 1 6 4 ' 2( 1 6 1 6 1 2 3 5 2 ) 9 16 % 9 ) 9 ' * 4 ' 2 ' * 4 '* 4 3 6 59 3 9 5 + 5 2 % 6 ' * * 41 4 ,- 4 4 5 4 3 . /' 1 9 ,2 ' * % 49 5 4 ' ' 0 4 6 4 9 4 '- 2 '2 4 1 4 '* 4 3 6 5 2 ' * 4 ' 9 3 1 40 4 ' * 6 ) 2 ' 1 9 2 ' * 4 '2 9 ' * 5 6 - 2 '2 4 1 4 '- 36 ' 3 ,6 4. 44 3 6 2 9 ' * 5 6 - 36 ' 3 ,6 40 4 ' * 6 % 4 5 6 4 ' 9 3 1 4 4 - 4 16 * 4 ,) 4 3 4 ' 2 ' * 4 '2 ) 6 %% 49 5 . 56 5 4 4 '- 446 3 ' 1 3 % 6 4 1 4 5 + ( 1 6 1 6 0 4 ' * 6 1 9 ' 7 9 444 %8 9 + : ; + 8 + = ; + 8 + ( ; 4 '8 9 + ; . ( 1 6 1 6 6 ' 64 - 4 1 62 4 1 62 ' * 4 '2 9 ' * 5 6 - 36 ' 3 ,- 4 ' * 4 1 : 0 4 6 1 9? @6 4' 6 4 6 A 4 '0 4 3 6 ( 1 6 1 6 6 ,4 5 9 4 ' 2 44 ,- 2 3 5 4 ,4 4 ' 6 4 1 4 5 4 4 ' 6 - 2 3 32 %,32 - 2 3 5 4 ,4 4 '5 6 ,91 4 '5 2 ) 4 * 4 6 ) 2 3 6 9 1 ? B6 1 6 C : D 4 E ( ) E 9 6 E B6 1 6 9 C = D : ( : 4 E ( F ) E 6 E B6 1 6 : C ( D G 9 4 E = ( ) E 6 E B6 1 6 ( C D 9 4 E ( ( ) E 9 6 E