TUGAS 1 METODE NUMERIK & KOMPUTASI

Nama: ZELVIA ZAHARA RAMBENPM: 148110018

Tentukan persamaan garis dari data berikut :

n 1 2 3 4 5 6 7 8 9 10x 4 6 8 10 14 16 20 22 24 28 152y 30 18 22 28 14 22 16 8 20 8 186

Penyelesaian :Metode Kuadrat Terkecil

n 1 2 3 4 5 6 7 8 9 10x 4 6 8 10 14 16 20 22 24 28 152y 30 18 22 28 14 22 16 8 20 8 186xi.yi 120 108 176 280 196 352 320 176 480 224 2432

16 36 64 100 196 256 400 484 576 784 2912

18.6

15.2

b = 10(2432)-(152x186)10(2912) -(152)^2

b= -0.65691

a = 28.58511

y= 28,585 - 0,657 x

Maka Kurva Linear dari soal diatas dapat di gambarkan sebagai berikut :

x y X Y X Y1 27.928 11 21.358 21 14.7882 27.271 12 20.701 22 14.1313 26.614 13 20.044 23 13.4744 25.957 14 19.387 24 12.8175 25.3 15 18.73 25 12.166 24.643 16 18.073 26 11.503

xi2

y = (y)/n=

x= (x)/n=

b={(n(xiyi - xy))/(n((xi)^2 - (yi^2 )}

a=y-bx

= ( ) = /= ( ) = /

= ( )/( ^2 ^2 )

=

TUGAS 1 METODE NUMERIK & KOMPUTASI7 23.986 17 17.416 27 10.8468 23.329 18 16.759 28 10.1899 22.672 19 16.102 29 9.532

10 22.015 20 15.445 30 8.875

0 5 10 15 20 25 300

5

10

15

20

25

30

35dataLinear (data)y = 28,585 - 0,596 xLinear (y = 28,585 - 0,596 x)

METODE KUADRAT TERKECIL

TUGAS 2 METODE NUMERIK & KOMPUTASI

Nama: ZELVIA ZAHARA RAMBENPM: 148110018

PERBAIKAN

n 1 2 3 4 5 6 7 8 9x 2.5 3.5 5 6 7.5 10 12.5 15 17.5y 5 3.4 2 1.6 1.2 0.8 0.6 0.4 0.3

Gambarkan Kurva dari hasil :1) Metode Kuadrat Terkecil2) Polynominal

Penyelesaian :1) Metode Kuadrat Terkecil

n xi yi qi = log x pi = log yi qi. Pi qi21 2.5 5 0.39794 0.69897 0.27815 0.158362 3.5 3.4 0.54407 0.53148 0.28916 0.296013 5 2 0.69897 0.30103 0.21041 0.488564 6 1.6 0.77815 0.20412 0.15884 0.605525 7.5 1.2 0.87506 0.07918 0.06929 0.765736 10 0.8 1 -0.09691 -0.09691 17 12.5 0.6 1.09691 -0.22185 -0.24335 1.203218 15 0.4 1.17609 -0.39794 -0.46801 1.383199 17.5 0.3 1.24304 -0.52288 -0.64996 1.54514

10 20 0.3 1.30103 -0.52288 -0.68028 1.69268 9.11126 0.05232 -1.13267 9.13840

B=B= -1.4103784039

0.005232388

q= 0.911125989

A= 1.2902648062

a) y= a0 + a1x + a2x2

b) y= a0 + a1x + a2x2 + a3x

3

(n.qi.pi)-(qi x pi)/(n.qi2)-(qi)2

p= (log yi )/np=

q= (log xi )/n

A= p - B.q

P= 1.290264806 -1.41 q

A= log aa= 19.51034

maka persamaan :y=

Maka Kurva dari soal diatas dapat di gambarkan sebagai berikut :y=

x y x y x y1 19.51034 11 0.662990163 21 0.2663393282 7.340043486 12 0.586422891 22 0.2494255163 4.143270845 13 0.523821344 23 0.2342681814 2.761419759 14 0.471835494 24 0.2206199135 2.015824126 15 0.428086098 25 0.2082765816 1.558752343 16 0.3908409127 1.254171169 17 0.3588114168 1.038882004 18 0.3310210459 0.879876686 19 0.306717372

10 0.758379236 20 0.285312125

19,51034 x -1,41038

19,51034 x -1,41038

0 5 10 15 20 250

5

10

15

20

25

Metode Kuadrat Terkecil

Data Column C

TUGAS 2 METODE NUMERIK & KOMPUTASI contoh:

10 n 1 2 3 4 520 99.5 x 1 2 3 4 5

0.3 15.6 y 0.5 1.7 3.4 5.7 8.4

n xi yi qi = log x pi = log yi1 1 0.5 0.00000 -0.301032 2 1.7 0.30103 0.230453 3 3.4 0.47712 0.531484 4 5.7 0.60206 0.755875 5 8.4 0.69897 0.92428

2.07918 2.14105

B= 1.751723648P= 0.428

qi2 q= 0.4160.15836 A= -0.3002197950.29601 a= 0.5010.488560.60552 y= 0.4980.76573

11.20321 x y1.38319 1 0.4981.54514 2 1.6851.69268 3 3.4359.13840 4 5.695

5 8.43

6 11.61

7 15.23

8 19.25

0 5 10 15 20 250

5

10

15

20

25

Metode Kuadrat Terkecil

Data Column C

= B .

= (log ) /

pi = log yi qi. Pi qi2-0.30103 0.00000 0.000000.23045 0.06937 0.090620.53148 0.25358 0.227640.75587 0.45508 0.362480.92428 0.64604 0.488562.14105 1.42408 1.16930

-3.028 -1.9872376211.523500597

2) Polynominal a. y= a0 + a1x + a2x2

n xi yi xi.yi1 2.5 5 6.25 15.625 39.0625 12.52 3.5 3.4 12.25 42.875 150.0625 11.93 5 2 25 125 625 104 6 1.6 36 216 1296 9.65 7.5 1.2 56.25 421.875 3164.063 96 10 0.8 100 1000 10000 87 12.5 0.6 156.25 1953.125 24414.06 7.58 15 0.4 225 3375 50625 69 17.5 0.3 306.25 5359.375 93789.06 5.25

10 20 0.3 400 8000 160000 6 99.5 15.6 1323.25 20508.88 344102 85.75

10 99.5 1323.25 15.699.5 1323.25 20508.88 = 85.75

1323.25 20508.875 344102.31 723.625

metode eliminasi gauss jordan

1 9.95 132.33 1.560 333.225 7342.54 = -69.470 7342.5375 169003.2563 -1340.645

1 0 -86.92100 3.634350 1 22.03477 = -0.208480 0 7212.103309 190.11074

1 0 0 5.925590 1 0 = -0.789310 0 1 0.02636

maka diperoleh := 5.92559= -0.78931= 0.02636

persamaan kurva diperoleh :

xi2 xi3 xi4

a0a1a2

a0a1a2

a0a1a2

a0a1a2

a0a1a2

y=

Maka Kurva dari soal diatas dapat di gambarkan sebagai berikut :y=

5,92559 - 0,78931 x + 0,02636 x2

5,92559 - 0,78931 x + 0,02636 x2

x y x y-3 8.53076 11 0.43274-2 7.60965 12 0.24971-1 6.74126 13 0.11940 5.92559 14 0.041811 5.16264 15 0.016942 4.45241 16 0.044793 3.7949 17 0.125364 3.19011 18 0.258655 2.63804 19 0.444666 2.13869 20 0.683397 1.69206 21 0.974848 1.29815 22 1.319019 0.95696 23 1.7159

10 0.66849 24 2.16551

-5 0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

Datay= a0 + a1x + a2x2

x

y

31.2541.65

5057.667.580

93.7590

91.875120

723.63

15.685.75

723.625

1.56-69.47

-1340.645

3.63435-0.20848

190.11074

5.92559-0.789310.02636

xi2.yi

-5 0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

Datay= a0 + a1x + a2x2

x

y

2) Polynominal b. y = a0 + a1x + a2x2 + a3x3

n xi yi1 2.5 5 6.25 15.625 39.06252 3.5 3.4 12.25 42.875 150.06253 5 2 25 125 6254 6 1.6 36 216 12965 7.5 1.2 56.25 421.875 3164.0636 10 0.8 100 1000 100007 12.5 0.6 156.25 1953.125 24414.068 15 0.4 225 3375 506259 17.5 0.3 306.25 5359.375 93789.06

10 20 0.3 400 8000 160000 99.5 15.6 1323.25 20508.88 344102

10 99.5 1323.25 20508.8899.5 1323.25 20508.88 344102.31

1323.25 20508.875 344102.31 6041113.7220508.875 344102.313 6041113.72 109170564.6

metode eliminasi gauss jordan 1 9.950 132.32500 2050.8880 333.225 7342.53750 140039.0060 7342.538 169003.26 3327276.8340 140039.006 3327276.83 67109169.202

1 0 -86.92100 -2130.635830 1 22.03477 420.253600 0 7212.10331 241549.008270 0 241549.01 8257272.5207

1 0 0 780.537270 1 0 -317.738840 0 1 33.492170 0 0 167271.463

1 0 0 00 1 0 00 0 1 0

xi2 xi3 xi4

a0a1a2a3

a0a1a2a3

a0a1a2a3

a0a1a2a3

a0a1a2

0 0 0 1

maka diperoleh := 8.30825= -1.75924= 0.12860= -0.00305

a3

a0a1a2a3

persamaan kurva diperoleh :y=

Maka Kurva dari soal diatas dapat di gambarkan sebagai berikut :

x y x y-3 14.82572 11 0.45766-2 12.36553 12 0.44537-1 10.19914 13 0.470680 8.30825 14 0.515291 6.67456 15 0.56092 5.27977 16 0.589213 4.10558 17 0.581924 3.13369 18 0.520735 2.3458 19 0.387346 1.72361 20 0.163457 1.24882 21 -0.169248 0.90313 22 -0.629039 0.66824 23 -1.23422

10 0.52585

8,30825 - 1,75924x + 0,12860 x2 - 0,00305x3

-5 0 5 10 15 20 250

2

4

6

8

10

12

14

16

Data

y=a0 + a1x1 + a2x2 + a3x3

x

y

xi.yi97.65625 244.140625 12.5 31.25 78.125525.2188 1838.26563 11.9 41.65 145.775

3125 15625 10 50 2507776 46656 9.6 57.6 345.6

23730.47 177978.516 9 67.5 506.25100000 1000000 8 80 800

305175.8 3814697.27 7.5 93.75 1171.875759375 11390625 6 90 1350

1641309 28722900 5.25 91.875 1607.8133200000 64000000 6 120 2400

6041114 109170565 85.75 723.625 8655.438

15.6= 85.75

723.6258655.4375

1.56= -69.47

-1340.645-23338.408

3.63435366= -0.2084778

190.1107365856.61017

5.9255875= -0.7893135

0.02635996-510.61139

8.30824871= -1.7592403

0.12859787

xi5 xi6 xi2.yi xi3.yi

-0.0030526

-5 0 5 10 15 20 250

2

4

6

8

10

12

14

16

Data

y=a0 + a1x1 + a2x2 + a3x3

x

y

TUGAS NUMERIK 4

Diketahui data uji baja sebagai berikut:

Load (KN) Strain (mm/mm)0 16.593 0.0002181 22.777 0.0002932 34.365 0.0004443 40.479 0.0005214 45.844 0.000592

Tentukan regangan baja pada saan beban = 28,821 KN

Penyelesaian :Interpolasi Lagrange berderajat 4

p (x) = Y0 (x-x1)(x-x2)(x-x3)(x-x4) + Y1 (x-x0)(x-x2)(x-x3)(x-x4)(x0-x1)(x0-x2)(x0-x3)(x0-X4) (x1-x0)(x1-x2)(x1-x3)(x1-X4)

+ Y2 + Y3 (x-x0)(x-x1)(x-x2)(x-x4)(x2-x0)(x2-x1)(x2-x3)(x2-X4) (x3-x0)(x3-x1)(x3-x2)(x3-x4)

+ Y4 (x-x0)(x-x1)(x-x2)(x-x3)(x4-x0)(x4-x1)(x4-x2)(x4-x3)

p (x) = (0,000218) (x-22,777)(x-34,365)(x-40,479)(x-45,844) +(0,000293) (x-16,593)(x-34,365)(x-40,479)(x-45,844)(16,593-22,777)(16,593-34,365)(16,593-40,479)(16,593-45,844) (22,777-16,593)(22,777-34,365)(22,777-40,479)(22,777-45,844)

+(0,000444) (x-16,593)(x-22,777)(x-40,479)(x-45,844) +(0,000521) (x-16,593)(x-22,777)(x-34,365)(x-45,844)(34,365-16,593)(34,365-22,777)(34,365-40,479)(34,365-45,844) (40,479-16,593)(40,479-22,777)(40,479-34,365)(40,479-45,844)

+(0,000592) (x-16,593)(x-22,777)(x-34,365)(x-40,479)(45,844-16,593)(45,844-22,777)(45,844-34,365)(45,844-40,479)

P(x)= 0.000218 -6649.7884210074 + 0.000293 -13453.6090026603 +76787.3944372625 -29261.1519561905

0.000444 14666.9575779363 + 0.000521 6974.9196098884 +14453.5417531652 -13869.5038378549

0.000592 4776.691112734541553.3092040302

P(x)= -1.8879E-05 + 1.3471E-04 + 4.5056E-04 + -2.6201E-04 + 6.8052E-05

P(28,821)= 0.000372

(x-x0)(x-x1)(x-x3)(x-x4)

Metode Kuadran Terkecilkuadrat terkecil kurvapolynominalinterpolasi polinom logrange