Metode Numerik

1. UTS Metode Numerik
Tekmnik Informatika
Hari Santoso
06550023
1). f(x)=cos2px+(2x+q), maka jika p dan q disesuaikan NIM menjadi cos(2*2*x)+(2*x+3) atau cos(4x)+(2x+3)
Catatan khusus pekerjaan saya :
Toleransi Lebar Selang = 0.0000001 (epsilon 1)
Nilai Hampiran Akar = 0.00000000001 (epsilon 2); f(x) mendekati 0
A. Hampiran akar
– Menggunakan metode Biseksi (Bagi Dua) hasilnya : -1.8016114476295115
dengan dua titik awal -3 dan 2
– Menggunakan metode Regula Falsi : -1.8016114476286291
dengan dua titik awal -3 dan 2
– Menggunakan metode Newton Raphson : -1.801611447628629
dengan titik awal -2
– Menggunakan metode Secant : -1.8016114476267557
dengan dua titik awal -3 dan 2
B. Galat Relatif (εRA) dari Xn (hampiran akar ke n)
– Metode Biseksi
n Xn X(n+1) (εRA) = ((X(n+1))-X)/X(n+1)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
0.0
-0.5
-1.75
-2.375
-2.0625
-1.90625
-1.828125
-1.7890625
-1.80859375
-1.798828125
-1.8037109375
-1.80126953125
-1.802490234375
-1.8018798828125
-1.80157470703125
-1.801727294921875
-1.8016510009765625
-1.8016128540039062
-1.8015937805175781
-1.8016033172607422
-1.8016080856323242
-1.8016104698181152
-1.8016116619110107
-1.801611065864563
-1.8016113638877869
-1.8016115128993988
-1.8016114383935928
-1.8016114756464958
-1.8016114570200443
-1.8016114477068186
-1.8016114430502057
-1.8016114453785121
-1.8016114465426654
-1.801611447124742
-1.8016114474157803
-1.8016114475612994
-1.801611447634059
-1.8016114475976792
-0.5
-1.75
-2.375
-2.0625
-1.90625
-1.828125
-1.7890625
-1.80859375
-1.798828125
-1.8037109375
-1.80126953125
-1.802490234375
-1.8018798828125
-1.80157470703125
-1.801727294921875
-1.8016510009765625
-1.8016128540039062
-1.8015937805175781
-1.8016033172607422
-1.8016080856323242
-1.8016104698181152
-1.8016116619110107
-1.801611065864563
-1.8016113638877869
-1.8016115128993988
-1.8016114383935928
-1.8016114756464958
-1.8016114570200443
-1.8016114477068186
-1.8016114430502057
-1.8016114453785121
-1.8016114465426654
-1.801611447124742
-1.8016114474157803
-1.8016114475612994
-1.801611447634059
-1.8016114475976792
-1.801611447615869
1.0
0.7142857142857143
0.2631578947368421
0.15151515151515152
0.08196721311475409
0.042735042735042736
0.021834061135371178
0.01079913606911447
0.0054288816503800215
0.0027070925825663237
0.0013553808620222283
6.772314777190844E-4
3.3873043831718716E-4
1.6939390859504693E-4
8.46897813309846E-5
4.234668383118918E-5
2.1173790235494895E-5
1.0587007201282299E-5
5.293475579609109E-6
2.6467307846021674E-6
1.3233636410074397E-6
6.616813826811779E-7
3.308408007961882E-7
1.6542037303418978E-7
8.27101796761205E-8
4.135509154830378E-8
2.0677545346590998E-8
1.033877278018572E-8
5.169386416815416E-9
2.584693215088347E-9
1.2923466058740138E-9
6.461733025194669E-10
3.230866511553485E-10
1.61543325551578E-10
8.077166276926494E-11
4.038583138300145E-11
2.019291569190848E-11
1.0096457845852301E-11

2. 39
40
-1.801611447615869
-1.801611447624964
-1.801611447624964
-1.8016114476295115
5.0482289229006665E-12
2.524114461443962E-12
– Metode Regula Falsi
n Xn X(n+1) (εRA) = ((X(n+1))-X)/X(n+1)
1
2
3
4
5
6
7
8
0.0
-1.8035563490771485
-1.797953034224506
-1.801604801477325
-1.8016114356729023
-1.8016114476071223
-1.8016114476285903
-1.801611447628629
-1.8035563490771485
-1.797953034224506
-1.801604801477325
-1.8016114356729023
-1.8016114476071223
-1.8016114476285903
-1.801611447628629
-1.8016114476286291
1.0
0.0031164967860571793
0.0020269524425248187
3.6823675993484088E-6
6.624191954642433E-9
1.191596476933161E-11
2.144511310016917E-14
1.2324777643775381E-16
– Metode Newton Raphson
n Xn X(n+1) (εRA) = ((X(n+1))-X)/X(n+1)
1
2
3
4
-2.0
-1.807719189790674
-1.8016450051978534
-1.8016114486748949
-1.807719189790674
-1.8016450051978534
-1.8016114486748949
-1.801611447628629
0.10636652600429128
0.0033714658411041643
1.8625837987006956E-5
5.807389624069679E-10
– Metode Secant
n Xn X(n+1) (εRA) = ((X(n+1))-X)/X(n+1)
1
2
3
4
5
6
7
8
9
10
11
12
2.0
-3.8035563490771485
-1.2231320622744302
-1.527154917782535
-0.09544665943497099
-2.0018169103475882
-1.5513403115519542
-1.7477976253246201
-1.8301133003465275
-1.8000848729605656
-1.801573851656067
-1.801611501216167
-3.8035563490771485
-1.2231320622744302
-1.527154917782535
-0.09544665943497099
-2.0018169103475882
-1.5513403115519542
-1.7477976253246201
-1.8301133003465275
-1.8000848729605656
-1.801573851656067
-1.801611501216167
-1.8016114476267557
1.5258236809046504
2.109685753805182
0.19907794027180503
15.000087659673461
0.9523199854384296
0.2903789680711508
0.11240278103489112
0.04497845844097255
0.016681673090544174
8.264877368933529E-4
2.0897713005595138E-5
2.9745265786088064E-8

3. C. Grafik Konvergensi Akar
Bentuk Grafik dalam Aplikasi
Grafik Konvergensi Akar
Dari grafik, bisa diperhatikan konvergensinya terjadi di sekita -1,8. Pada daerah itu, nilainya sudah mengalami
kekontinuan yang digambarkan dengan garis lurus. Pada grafik konvergensi di atas, metode yang digunakan
yaitu Biseksi. Jadi untuk metode yang lain, tentu akan sedikit berbeda, tapi konvergensinya tetap pada sekitar
titik -1,8 seperti yang telah dicari sebelumnya. Nah biar lebih jelas, silakan perhatikan tabel pada bagian D.

4. D. Analisa Kecepatan Iterasi mencari akar dengan Java
– Metode Biseksi (Bagi Dua)
r a c b f(a) f(c) f(b) Lebar
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
-3.0
-3.0
-3.0
-2.375
-2.0625
-1.90625
-1.828125
-1.828125
-1.80859375
-1.80859375
-1.8037109375
-1.8037109375
-1.802490234375
-1.8018798828125
-1.8018798828125
-1.801727294921875
-1.8016510009765625
-1.8016128540039062
-1.8016128540039062
-1.8016128540039062
-1.8016128540039062
-1.8016128540039062
-1.8016116619110107
-1.8016116619110107
-1.8016116619110107
-1.8016115128993988
-1.8016115128993988
-1.8016114756464958
-1.8016114570200443
-1.8016114477068186
-1.8016114477068186
-1.8016114477068186
-1.8016114477068186
-1.8016114477068186
-1.8016114477068186
-1.8016114477068186
-1.801611447634059
-1.801611447634059
-1.801611447634059
-1.801611447634059
-0.5
-1.75
-2.375
-2.0625
-1.90625
-1.828125
-1.7890625
-1.80859375
-1.798828125
-1.8037109375
-1.80126953125
-1.802490234375
-1.8018798828125
-1.80157470703125
-1.801727294921875
-1.8016510009765625
-1.8016128540039062
-1.8015937805175781
-1.8016033172607422
-1.8016080856323242
-1.8016104698181152
-1.8016116619110107
-1.801611065864563
-1.8016113638877869
-1.8016115128993988
-1.8016114383935928
-1.8016114756464958
-1.8016114570200443
-1.8016114477068186
-1.8016114430502057
-1.8016114453785121
-1.8016114465426654
-1.801611447124742
-1.8016114474157803
-1.8016114475612994
-1.801611447634059
-1.8016114475976792
-1.801611447615869
-1.801611447624964
-1.8016114476295115
2.0
-0.5
-1.75
-1.75
-1.75
-1.75
-1.75
-1.7890625
-1.7890625
-1.798828125
-1.798828125
-1.80126953125
-1.80126953125
-1.80126953125
-1.80157470703125
-1.80157470703125
-1.80157470703125
-1.80157470703125
-1.8015937805175781
-1.8016033172607422
-1.8016080856323242
-1.8016104698181152
-1.8016104698181152
-1.801611065864563
-1.8016113638877869
-1.8016113638877869
-1.8016114383935928
-1.8016114383935928
-1.8016114383935928
-1.8016114383935928
-1.8016114430502057
-1.8016114453785121
-1.8016114465426654
-1.801611447124742
-1.8016114474157803
-1.8016114475612994
-1.8016114475612994
-1.8016114475976792
-1.801611447615869
-1.801611447624964
-2.1561460412675078
-2.1561460412675078
-2.1561460412675078
-2.7471721561963784
-1.510747937452222
-0.5855141432845307
-0.14084376281497168
-0.14084376281497168
-0.03647253654195837
-0.03647253654195837
-0.010918155089783221
-0.010918155089783221
-0.004564879615149953
-0.0013936041850319825
-0.0013936041850319825
-6.01345870745118E-4
-2.0530091988679722E-4
-7.299505949576179E-6
-7.299505949576179E-6
-7.299505949576179E-6
-7.299505949576179E-6
-7.299505949576179E-6
-1.1121880566511422E-6
-1.1121880566511422E-6
-1.1121880566511422E-6
-3.387742844029873E-7
-3.387742844029873E-7
-1.454208748974395E-7
-4.87441751406692E-8
-4.058264835293812E-10
-4.058264835293812E-10
-4.058264835293812E-10
-4.058264835293812E-10
-4.058264835293812E-10
-4.058264835293812E-10
-4.058264835293812E-10
-2.818312250241206E-11
-2.818312250241206E-11
-2.818312250241206E-11
-2.818312250241206E-11
1.5838531634528576
0.2539022543433046
-2.7471721561963784
-1.510747937452222
-0.5855141432845307
-0.14084376281497168
0.06435608928769787
-0.03647253654195837
0.014408685484931905
-0.010918155089783221
0.0017740808647616069
-0.004564879615149953
-0.0013936041850319825
1.906878632146336E-4
-6.01345870745118E-4
-2.0530091988679722E-4
-7.299505949576179E-6
9.169593440627732E-5
4.2198653149583265E-5
1.7449683327619425E-5
5.075116120578649E-6
-1.1121880566511422E-6
1.981465746370148E-6
4.3463927346110154E-7
-3.387742844029873E-7
4.793252128543202E-8
-1.454208748974395E-7
-4.87441751406692E-8
-4.058264835293812E-10
2.376334751197362E-8
1.167876051422212E-8
5.63646707085752E-9
2.6153202936640696E-9
1.104746849556193E-9
3.494602385245571E-10
-2.818312250241206E-11
1.606385024999213E-10
6.622768999875461E-11
1.9022339259322507E-11
-4.580447132696008E-12
6.854499966191386
1.5838531634528576
0.2539022543433046
0.2539022543433046
0.2539022543433046
0.2539022543433046
0.2539022543433046
0.06435608928769787
0.06435608928769787
0.014408685484931905
0.014408685484931905
0.0017740808647616069
0.0017740808647616069
0.0017740808647616069
1.906878632146336E-4
1.906878632146336E-4
1.906878632146336E-4
1.906878632146336E-4
9.169593440627732E-5
4.2198653149583265E-5
1.7449683327619425E-5
5.075116120578649E-6
5.075116120578649E-6
1.981465746370148E-6
4.3463927346110154E-7
4.3463927346110154E-7
4.793252128543202E-8
4.793252128543202E-8
4.793252128543202E-8
4.793252128543202E-8
2.376334751197362E-8
1.167876051422212E-8
5.63646707085752E-9
2.6153202936640696E-9
1.104746849556193E-9
3.494602385245571E-10
3.494602385245571E-10
1.606385024999213E-10
6.622768999875461E-11
1.9022339259322507E-11
5.0
2.5
1.25
0.625
0.3125
0.15625
0.078125
0.0390625
0.01953125
0.009765625
0.0048828125
0.00244140625
0.001220703125
6.103515625E-4
3.0517578125E-4
1.52587890625E-4
7.62939453125E-5
3.814697265625E-5
1.9073486328125E-5
9.5367431640625E-6
4.76837158203125E-6
2.384185791015625E-6
1.1920928955078125E-6
5.960464477539062E-7
2.980232238769531E-7
1.4901161193847656E-7
7.450580596923828E-8
3.725290298461914E-8
1.862645149230957E-8
9.313225746154785E-9
4.6566128730773926E-9
2.3283064365386963E-9
1.1641532182693481E-9
5.820766091346741E-10
2.9103830456733704E-10
1.4551915228366852E-10
7.275957614183426E-11
3.637978807091713E-11
1.8189894035458565E-11
9.094947017729282E-12
– Metode Regula Falsi
r a c b f(a) f(c) f(b) Lebar
1
2
3
4
5
6
7
8
-3.0
-1.8035563490771485
-1.8035563490771485
-1.8035563490771485
-1.8035563490771485
-1.8035563490771485
-1.8035563490771485
-1.8035563490771485
-1.8035563490771485
-1.797953034224506
-1.801604801477325
-1.8016114356729023
-1.8016114476071223
-1.8016114476285903
-1.801611447628629
-1.8016114476286291
2.0
2.0
-1.797953034224506
-1.801604801477325
-1.8016114356729023
-1.8016114476071223
-1.8016114476285903
-1.801611447628629
-2.1561460412675078
-0.010112795266789765
-0.010112795266789765
-0.010112795266789765
-0.010112795266789765
-0.010112795266789765
-0.010112795266789765
-0.010112795266789765
-0.010112795266789765
0.01892322513325584
3.449524358623002E-5
6.205369573741848E-8
1.1162604174330681E-10
2.0117241206207837E-13
6.661338147750939E-16
-4.440892098500626E-16
6.854499966191386
6.854499966191386
0.01892322513325584
3.449524358623002E-5
6.205369573741848E-8
1.1162604174330681E-10
2.0117241206207837E-13
6.661338147750939E-16
5.0
3.8035563490771485
0.005603314852642427
0.001951547599823522
0.0019449134042461846
0.001944901470026128
0.0019449014485581895
0.0019449014485195537
– Metode Newton Raphson
r Xr Xr+1 Xr+1 - Xr
1
2
3
4
-2.0
-1.807719189790674
-1.8016450051978534
-1.8016114486748949
0.0
-2.0
-1.807719189790674
-1.8016450051978534
2.0
0.19228081020932608
0.006074184592820497
3.355652295855549E-5
– Metode Secant
r Xr Xr+1 Xr+1 - Xr
1
2
3
4
5
6
7
8
9
10
11
12
2.0
-3.8035563490771485
-1.2231320622744302
-1.527154917782535
-0.09544665943497099
-2.0018169103475882
-1.5513403115519542
-1.7477976253246201
-1.8301133003465275
-1.8000848729605656
-1.801573851656067
-1.801611501216167
-3.8035563490771485
-1.2231320622744302
-1.527154917782535
-0.09544665943497099
-2.0018169103475882
-1.5513403115519542
-1.7477976253246201
-1.8301133003465275
-1.8000848729605656
-1.801573851656067
-1.801611501216167
-1.8016114476267557
5.803556349077148
2.5804242868027183
0.3040228555081048
1.431708258347564
1.9063702509126172
0.45047659879563406
0.19645731377266595
0.0823156750219074
0.030028427385961898
0.0014889786955014639
3.764956009999487E-5
5.358941135291673E-8
Jadi dalam kasus ini, metode Newton Raphson memiliki kecepatan menghapiri akar yang paling cepat
dibanding metode yang lain. Tapi dalam kasus lain, bisa saja metode Newton Raphson memiliki kelemahan
yang tidak bisa dipecahkan terutama rumus-rumus yang rumit. Dan hal ini kemudian ada metode Secant
sebagai modifikasi dari Newton Raphson yang bisa memperbaiki Newton Raphson.

5. 2). Lagrange
x 1,0 1,3 1,6 1,9 2,2
y 0,7652 0,6201 0,4554 0,2818 0,1104
Hitung f(1.5p) → p= angka nim trakhir, maka f(1.53)
code here
8<---------------------------------------------------------------
class lagrange{
public static void main (String haripinter[]){
int i,j;
double a=1, b=0, z=1.53, c=0;
double[] x={1,1.3,1.6,1.9,2.2};
double[] y={0.7652,0.6201,0.4554,0.2818,0.1104};
for (i=0;i<x.length ;i++ ){
a=1;
for (j=0;j<y.length ;j++ ){
if(i!=j){
a=a*(z-x[j])/(x[i]-x[j]);
}
}
b=a*y[i];
c=c+b;
}
System.out.println("+--------------------+");
System.out.println("| "+c+" |");
System.out.println("+--------------------+");
}
}
8<----------------------------------------------------------------
Output :
+--------------------+
| 0.4950260080452676 |
+--------------------+