METODE NUMERIK METODE ITERASI GAUSS SEID
METODE NUMERIK
“METODE ITERASI GAUSS-SEIDEL DAN
DIFERENSIASI NUMERIK”
Nama
: Rian Adi Wirawan
NIM
: 21060112130134
Kelas
:C
JURUSAN TEKNIK ELEKTRO
FAKULTAS TEKNIK
UNIVERSITAS DIPONEGORO
SEMARANG
METODE ITERASI GAUSS-SEIDEL
PR :
x+ y+ 2 z=9
2 x + 4 y−3 z=1
3 x+6 y −5 z=0
Misalkan :
x=9− y−2 z
y=
1−2 x+ 3 z
4
z=
3 x +6 y
5
Inisialisasi :
y=1 ; z=1
Iterasi x
y
z
error x error y error z
0
1
1
1
6
-2
1.2
-1.5
0.1
2
8.6
-3.15
1.38
1.3
-0.575
0.09
3
9.39
-3.41
1.542
0.395
-0.13
0.081
4
9.326 -3.2565
1.6878
-0.032 0.07675
0.0729
5
8.8809 -2.9246 1.81902 0.22255 0.16595 0.06561
- 1.93711
- 0.19779 0.05904
6 8.28656 2.52902
8 0.29717
3
9
7.65477
- 2.04340
- 0.20223 0.05314
7
9 2.12455
6 0.31589
2
4
7.03773
- 2.13906
- 0.19411
8
9 1.73631
6 0.30852
8 0.04783
6.45818
- 2.22515
- 0.18076 0.04304
9
3 1.37479
9 0.28978
1
7
5.92447
- 2.30264
- 0.16571 0.03874
10
5 1.04337
3 0.26685
2
2
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
5.43808
2
4.99730
1
4.59908
5
4.23993
3
3.91631
8
3.62487
5
3.36248
2
3.12628
2
2.91367
7
2.72232
1
2.55009
5
2.39508
8
2.25558
1
2.13002
4
2.01702
2
1.91532
1.82378
8
1.74140
9
1.66726
8
1.60054
1
1.54048
7
1.48643
9
1.43779
0.74206
0.46937
0.22319
0.00125
0.19868
9
0.37872
6
0.54080
6
0.68670
2
2.37237
9
2.43514
1
2.49162
7
2.54246
4
2.58821
8
2.62939
6
2.66645
6
2.69981
1
0.81802
0.93621
2
1.04258
8
1.13832
7
1.22449
4
1.30204
4
1.37183
9
1.43465
5
2.72983
2.75684
7
2.78116
2
2.80304
6
2.82274
1
2.84046
7
-0.1063
0.09568
0.08611
2.85642
2.87077
8
2.88370
1
-0.0565
0.05085
0.04577
0.04119
0.03707
0.03336
0.03003
0.02702
-
1.49119
1.54207
1
1.58786
4
1.62907
7
1.66617
1.69955
3
1.72959
2.89533
2.90579
7
2.91521
8
2.92369
6
2.93132
6
2.93819
-0.2432
0.22039
0.19911
0.17958
0.16181
0.14572
-0.1312
-0.1181
-0.0775
0.06975
0.06278
0.15065
5
0.13634
6
0.12309
0.11097
0.09996
8
0.09001
8
0.08104
0.07294
8
0.06565
9
0.05909
6
0.05318
8
0.04787
0.04308
3
0.03877
5
0.03489
8
0.03140
8
0.02826
7
0.02544
0.02289
6
0.02060
7
0.01854
6
0.01669
2
0.01502
0.03486
8
0.03138
1
0.02824
3
0.02541
9
0.02287
7
0.02058
9
0.01853
0.01667
7
0.01500
9
0.01350
9
0.01215
8
0.01094
2
0.00984
8
0.00886
3
0.00797
7
0.00717
9
0.00646
1
0.00581
5
0.00523
3
0.00471
0.00423
9
0.00381
5
0.00343
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
5
1.39401
5
1.35461
4
1.31915
2
1.28723
7
1.25851
3
1.23266
2
1.20939
6
1.18845
6
1.16961
1
1.15265
1.13738
5
1.12364
6
1.11128
2
1.10015
3
1.09013
8
1.08112
4
Dari
7
1.75663
8
1.78097
4
1.80287
7
1.82258
9
1.84033
1.85629
7
1.87066
7
1.88360
1
1.89524
1.90571
6
1.91514
5
1.92363
1.93126
7
1.93814
1
1.94432
7
1.94989
4
table
4
2.94437
4
2.94993
7
2.95494
3
2.95944
9
2.96350
4
2.96715
4
2.97043
8
2.97339
4
2.97605
5
2.97844
9
2.98060
5
2.98254
4
2.98429
2.98586
1
2.98727
5
2.98854
7
0.02432
0.02189
-0.0197
0.01773
0.01596
0.01436
0.01293
0.01163
0.01047
0.00942
0.00848
0.00763
0.00687
0.00618
0.00556
0.00501
0.00451
diatas,
2
4
0.01352
0.01216
8
0.01095
1
0.00985
6
0.00887
1
0.00798
4
0.00718
5
0.00646
7
0.00309
0.00278
1
0.00250
3
0.00225
3
0.00202
8
0.00182
5
0.00164
2
0.00147
8
0.00582
0.00523
8
0.00471
4
0.00424
3
0.00381
8
0.00343
7
0.00309
3
0.00278
4
0.00133
0.00119
7
0.00107
8
didapat
x=1.081124 ≡ 1; y =1.949894 ≡2 ; z=2.988547 ≡3
error x=−0.00451; error y=0.002784 ; error z=0.000636 .
DIFERENSIASI NUMERIK
PR :
0.00097
0.00087
3
0.00078
6
0.00070
7
0.00063
6
bahwa
dengan
y=x 3−2 x 2−x
Metode Selisih Maju
x
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
f(x)
0
-0.05488
-0.119
-0.19163
-0.272
-0.35938
-0.453
-0.55213
-0.656
-0.76388
-0.875
-0.98863
-1.104
-1.22038
-1.337
-1.45313
-1.568
-1.68088
-1.791
-1.89763
-2
f'(x)
-1.0975
-1.2825
-1.4525
-1.6075
-1.7475
-1.8725
-1.9825
-2.0775
-2.1575
-2.2225
-2.2725
-2.3075
-2.3275
-2.3325
-2.3225
-2.2975
-2.2575
-2.2025
-2.1325
-2.0475
nilai
eksak
-1
-1.1925
-1.37
-1.5325
-1.68
-1.8125
-1.93
-2.0325
-2.12
-2.1925
-2.25
-2.2925
-2.32
-2.3325
-2.33
-2.3125
-2.28
-2.2325
-2.17
-2.0925
-2
error
-0.095
-0.0875
-0.08
-0.0725
-0.065
-0.0575
-0.05
-0.0425
-0.035
-0.0275
-0.02
-0.0125
-0.005
0.0025
0.01
0.0175
0.025
0.0325
0.04
0.0475
Rata-rata error = -0.02375
Metode Selisih Mundur
x
-0.05
f(x)
0.04487
5
nilai
eksak
f'(x)
-
-0.7925
error
-
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0
-0.05488
-0.119
-0.19163
-0.272
-0.35938
-0.453
-0.55213
-0.656
-0.76388
-0.875
-0.98863
-1.104
-1.22038
-1.337
-1.45313
-1.568
-1.68088
-1.791
-1.89763
-2
-0.8975
-1.0975
-1.2825
-1.4525
-1.6075
-1.7475
-1.8725
-1.9825
-2.0775
-2.1575
-2.2225
-2.2725
-2.3075
-2.3275
-2.3325
-2.3225
-2.2975
-2.2575
-2.2025
-2.1325
-2.0475
-1
-1.1925
-1.37
-1.5325
-1.68
-1.8125
-1.93
-2.0325
-2.12
-2.1925
-2.25
-2.2925
-2.32
-2.3325
-2.33
-2.3125
-2.28
-2.2325
-2.17
-2.0925
-2
-0.1025
-0.095
-0.0875
-0.08
-0.0725
-0.065
-0.0575
-0.05
-0.0425
-0.035
-0.0275
-0.02
-0.0125
-0.005
0.0025
0.01
0.0175
0.025
0.0325
0.04
0.0475
Rata-rata error = -0.0275
Metode Selisih Tengahan
x
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
f(x)
0.04487
5
0
-0.05488
-0.119
-0.19163
-0.272
-0.35938
-0.453
-0.55213
-0.656
-0.76388
f'(x)
-0.9975
-1.19
-1.3675
-1.53
-1.6775
-1.81
-1.9275
-2.03
-2.1175
-2.19
nilai
eksak
-0.7925
-1
-1.1925
-1.37
-1.5325
-1.68
-1.8125
-1.93
-2.0325
-2.12
-2.1925
error
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
-0.875
-0.98863
-1.104
-1.22038
-1.337
-1.45313
-1.568
-1.68088
-1.791
-1.89763
-2
-2.2475
-2.29
-2.3175
-2.33
-2.3275
-2.31
-2.2775
-2.23
-2.1675
-2.09
-
-2.25
-2.2925
-2.32
-2.3325
-2.33
-2.3125
-2.28
-2.2325
-2.17
-2.0925
-2
Rata-rata error = -0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-
“METODE ITERASI GAUSS-SEIDEL DAN
DIFERENSIASI NUMERIK”
Nama
: Rian Adi Wirawan
NIM
: 21060112130134
Kelas
:C
JURUSAN TEKNIK ELEKTRO
FAKULTAS TEKNIK
UNIVERSITAS DIPONEGORO
SEMARANG
METODE ITERASI GAUSS-SEIDEL
PR :
x+ y+ 2 z=9
2 x + 4 y−3 z=1
3 x+6 y −5 z=0
Misalkan :
x=9− y−2 z
y=
1−2 x+ 3 z
4
z=
3 x +6 y
5
Inisialisasi :
y=1 ; z=1
Iterasi x
y
z
error x error y error z
0
1
1
1
6
-2
1.2
-1.5
0.1
2
8.6
-3.15
1.38
1.3
-0.575
0.09
3
9.39
-3.41
1.542
0.395
-0.13
0.081
4
9.326 -3.2565
1.6878
-0.032 0.07675
0.0729
5
8.8809 -2.9246 1.81902 0.22255 0.16595 0.06561
- 1.93711
- 0.19779 0.05904
6 8.28656 2.52902
8 0.29717
3
9
7.65477
- 2.04340
- 0.20223 0.05314
7
9 2.12455
6 0.31589
2
4
7.03773
- 2.13906
- 0.19411
8
9 1.73631
6 0.30852
8 0.04783
6.45818
- 2.22515
- 0.18076 0.04304
9
3 1.37479
9 0.28978
1
7
5.92447
- 2.30264
- 0.16571 0.03874
10
5 1.04337
3 0.26685
2
2
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
5.43808
2
4.99730
1
4.59908
5
4.23993
3
3.91631
8
3.62487
5
3.36248
2
3.12628
2
2.91367
7
2.72232
1
2.55009
5
2.39508
8
2.25558
1
2.13002
4
2.01702
2
1.91532
1.82378
8
1.74140
9
1.66726
8
1.60054
1
1.54048
7
1.48643
9
1.43779
0.74206
0.46937
0.22319
0.00125
0.19868
9
0.37872
6
0.54080
6
0.68670
2
2.37237
9
2.43514
1
2.49162
7
2.54246
4
2.58821
8
2.62939
6
2.66645
6
2.69981
1
0.81802
0.93621
2
1.04258
8
1.13832
7
1.22449
4
1.30204
4
1.37183
9
1.43465
5
2.72983
2.75684
7
2.78116
2
2.80304
6
2.82274
1
2.84046
7
-0.1063
0.09568
0.08611
2.85642
2.87077
8
2.88370
1
-0.0565
0.05085
0.04577
0.04119
0.03707
0.03336
0.03003
0.02702
-
1.49119
1.54207
1
1.58786
4
1.62907
7
1.66617
1.69955
3
1.72959
2.89533
2.90579
7
2.91521
8
2.92369
6
2.93132
6
2.93819
-0.2432
0.22039
0.19911
0.17958
0.16181
0.14572
-0.1312
-0.1181
-0.0775
0.06975
0.06278
0.15065
5
0.13634
6
0.12309
0.11097
0.09996
8
0.09001
8
0.08104
0.07294
8
0.06565
9
0.05909
6
0.05318
8
0.04787
0.04308
3
0.03877
5
0.03489
8
0.03140
8
0.02826
7
0.02544
0.02289
6
0.02060
7
0.01854
6
0.01669
2
0.01502
0.03486
8
0.03138
1
0.02824
3
0.02541
9
0.02287
7
0.02058
9
0.01853
0.01667
7
0.01500
9
0.01350
9
0.01215
8
0.01094
2
0.00984
8
0.00886
3
0.00797
7
0.00717
9
0.00646
1
0.00581
5
0.00523
3
0.00471
0.00423
9
0.00381
5
0.00343
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
5
1.39401
5
1.35461
4
1.31915
2
1.28723
7
1.25851
3
1.23266
2
1.20939
6
1.18845
6
1.16961
1
1.15265
1.13738
5
1.12364
6
1.11128
2
1.10015
3
1.09013
8
1.08112
4
Dari
7
1.75663
8
1.78097
4
1.80287
7
1.82258
9
1.84033
1.85629
7
1.87066
7
1.88360
1
1.89524
1.90571
6
1.91514
5
1.92363
1.93126
7
1.93814
1
1.94432
7
1.94989
4
table
4
2.94437
4
2.94993
7
2.95494
3
2.95944
9
2.96350
4
2.96715
4
2.97043
8
2.97339
4
2.97605
5
2.97844
9
2.98060
5
2.98254
4
2.98429
2.98586
1
2.98727
5
2.98854
7
0.02432
0.02189
-0.0197
0.01773
0.01596
0.01436
0.01293
0.01163
0.01047
0.00942
0.00848
0.00763
0.00687
0.00618
0.00556
0.00501
0.00451
diatas,
2
4
0.01352
0.01216
8
0.01095
1
0.00985
6
0.00887
1
0.00798
4
0.00718
5
0.00646
7
0.00309
0.00278
1
0.00250
3
0.00225
3
0.00202
8
0.00182
5
0.00164
2
0.00147
8
0.00582
0.00523
8
0.00471
4
0.00424
3
0.00381
8
0.00343
7
0.00309
3
0.00278
4
0.00133
0.00119
7
0.00107
8
didapat
x=1.081124 ≡ 1; y =1.949894 ≡2 ; z=2.988547 ≡3
error x=−0.00451; error y=0.002784 ; error z=0.000636 .
DIFERENSIASI NUMERIK
PR :
0.00097
0.00087
3
0.00078
6
0.00070
7
0.00063
6
bahwa
dengan
y=x 3−2 x 2−x
Metode Selisih Maju
x
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
f(x)
0
-0.05488
-0.119
-0.19163
-0.272
-0.35938
-0.453
-0.55213
-0.656
-0.76388
-0.875
-0.98863
-1.104
-1.22038
-1.337
-1.45313
-1.568
-1.68088
-1.791
-1.89763
-2
f'(x)
-1.0975
-1.2825
-1.4525
-1.6075
-1.7475
-1.8725
-1.9825
-2.0775
-2.1575
-2.2225
-2.2725
-2.3075
-2.3275
-2.3325
-2.3225
-2.2975
-2.2575
-2.2025
-2.1325
-2.0475
nilai
eksak
-1
-1.1925
-1.37
-1.5325
-1.68
-1.8125
-1.93
-2.0325
-2.12
-2.1925
-2.25
-2.2925
-2.32
-2.3325
-2.33
-2.3125
-2.28
-2.2325
-2.17
-2.0925
-2
error
-0.095
-0.0875
-0.08
-0.0725
-0.065
-0.0575
-0.05
-0.0425
-0.035
-0.0275
-0.02
-0.0125
-0.005
0.0025
0.01
0.0175
0.025
0.0325
0.04
0.0475
Rata-rata error = -0.02375
Metode Selisih Mundur
x
-0.05
f(x)
0.04487
5
nilai
eksak
f'(x)
-
-0.7925
error
-
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0
-0.05488
-0.119
-0.19163
-0.272
-0.35938
-0.453
-0.55213
-0.656
-0.76388
-0.875
-0.98863
-1.104
-1.22038
-1.337
-1.45313
-1.568
-1.68088
-1.791
-1.89763
-2
-0.8975
-1.0975
-1.2825
-1.4525
-1.6075
-1.7475
-1.8725
-1.9825
-2.0775
-2.1575
-2.2225
-2.2725
-2.3075
-2.3275
-2.3325
-2.3225
-2.2975
-2.2575
-2.2025
-2.1325
-2.0475
-1
-1.1925
-1.37
-1.5325
-1.68
-1.8125
-1.93
-2.0325
-2.12
-2.1925
-2.25
-2.2925
-2.32
-2.3325
-2.33
-2.3125
-2.28
-2.2325
-2.17
-2.0925
-2
-0.1025
-0.095
-0.0875
-0.08
-0.0725
-0.065
-0.0575
-0.05
-0.0425
-0.035
-0.0275
-0.02
-0.0125
-0.005
0.0025
0.01
0.0175
0.025
0.0325
0.04
0.0475
Rata-rata error = -0.0275
Metode Selisih Tengahan
x
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
f(x)
0.04487
5
0
-0.05488
-0.119
-0.19163
-0.272
-0.35938
-0.453
-0.55213
-0.656
-0.76388
f'(x)
-0.9975
-1.19
-1.3675
-1.53
-1.6775
-1.81
-1.9275
-2.03
-2.1175
-2.19
nilai
eksak
-0.7925
-1
-1.1925
-1.37
-1.5325
-1.68
-1.8125
-1.93
-2.0325
-2.12
-2.1925
error
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
-0.875
-0.98863
-1.104
-1.22038
-1.337
-1.45313
-1.568
-1.68088
-1.791
-1.89763
-2
-2.2475
-2.29
-2.3175
-2.33
-2.3275
-2.31
-2.2775
-2.23
-2.1675
-2.09
-
-2.25
-2.2925
-2.32
-2.3325
-2.33
-2.3125
-2.28
-2.2325
-2.17
-2.0925
-2
Rata-rata error = -0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-0.0025
-