TUGAS 2 METODE NUMERIK & KOMPUTASI

Nama: ZELVIA ZAHARA RAMBENPM: 148110018

n 1 2 3 4 5 6 7 8 9 10x 2.5 3.5 5 6 7.5 10 12.5 15 17.5 20 99.5y 5 3.4 2 1.6 1.2 0.8 0.6 0.4 0.3 0.3 15.6

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

Penyelesaian :

1) Metode Kuadrat Terkecil

n 1 2 3 4 5 6 7 8 9 10x 2.5 3.5 5 6 7.5 10 12.5 15 17.5 20 99.5y 5 3.4 2 1.6 1.2 0.8 0.6 0.4 0.3 0.3 15.6xi.yi 12.5 11.9 10 9.6 9 8 7.5 6 5.25 6 85.75

6.25 12.25 25 36 56.25 100 156.3 225 306.3 400 1323.25

1.56

9.95

b = 10(85,75)-(15x99,5)10(1323) -(99,5)^2

b= -0.20848

a = 3.63435

y= 3,63453 - 0,20848 x

a) y= a0 + a1x + a2x2

b) y= a0 + a1x + a2x2 + a3x3

xi2

y ̅*= (∑y)/n=

x ̅�= (∑y)/n= b=(n∑〖xiyi - ∑xy〗)/(n∑〖〖xi〗^2 - ∑〖yi〗^2 〗)

a=y ̅�-bx ̅�

𝑦 ̅�= (∑▒𝑦)/𝑛= 𝑥 '= (∑ )/ = ▒𝑦 𝑛

𝑏=(𝑛∑▒ − ∑▒〖𝑥𝑖𝑦𝑖 𝑥𝑦〗 )/( ∑𝑛 ▒ 〖〖𝑥𝑖〗 ^2 − ∑▒ 〖𝑦𝑖〗 ^2 〗)

𝑎= '− '𝑦 𝑏𝑥

TUGAS 2 METODE NUMERIK & KOMPUTASI

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

x y-3 4.26-2 4.0515-1 3.8430 3.63451 3.42612 3.21763 3.00914 2.80065 2.59216 2.38377 2.17528 1.96679 1.7582

10 1.549711 1.341212 1.132813 0.924314 0.715815 0.507316 0.298817 0.090418 -0.118119 -0.326620 -0.535121 -0.743522 -0.95223 -1.160524 -1.36925 -1.5775

-5 0 5 10 15 20 25 30

-2

-1

0

1

2

3

4

5

6

Metode Kuadrat Terkecil

Data

y = 3,63453 - 0,20848 x

x

y

TUGAS 2 METODE NUMERIK & KOMPUTASI

TUGAS 2 METODE NUMERIK & KOMPUTASI

Nama: ZELVIA ZAHARA RAMBENPM: 148110018

PERBAIKAN

n 1 2 3 4 5 6 7 8 9 10x 2.5 3.5 5 6 7.5 10 12.5 15 17.5 20 99.5y 5 3.4 2 1.6 1.2 0.8 0.6 0.4 0.3 0.3 15.6

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= -1.41037840394616

0.005232388

q= 0.911125989

A= 1.29026480620269

a) y= a0 + a1x + a2x2

b) y= a0 + a1x + a2x2 + a3x3

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

p ̅�= (∑log yi )/np =

q ̅�= (∑log xi )/nA= 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 y1 19.510342 7.3400434863 4.1432708454 2.7614197595 2.0158241266 1.5587523437 1.2541711698 1.0388820049 0.879876686

10 0.75837923611 0.66299016312 0.58642289113 0.52382134414 0.47183549415 0.42808609816 0.39084091217 0.35881141618 0.33102104519 0.30671737220 0.28531212521 0.26633932822 0.24942551623 0.23426818124 0.22061991325 0.208276581

19,51034 x -1,41038

19,51034 x -1,41038

0 5 10 15 20 25 300

5

10

15

20

25

DataColumn C

x

y

contoh:

n 1 2 3 4 5x 1 2 3 4 5y 0.5 1.7 3.4 5.7 8.4

n xi yi qi = log x pi = log yi qi. Pi qi21 1 0.5 0.00000 -0.30103 0.00000 0.000002 2 1.7 0.30103 0.23045 0.06937 0.09062

3 3 3.4 0.47712 0.53148 0.25358 0.22764

4 4 5.7 0.60206 0.75587 0.45508 0.362485 5 8.4 0.69897 0.92428 0.64604 0.48856

∑ 2.07918 2.14105 1.42408 1.16930

B= 1.751723648 -3.028 -1.98723762P= 0.428 1.523500597q= 0.416A= -0.30021979a= 0.501

y= 0.498

x y1 0.4982 1.6853 3.4354 5.6955 8.43

6 11.61

7 15.23

8 19.25

0 5 10 15 20 25 300

5

10

15

20

25

DataColumn C

x

y

𝐴= 𝑝 ̅� −B.𝑞 ̅�𝑞 ̅�= (∑▒log 𝑥𝑖 )/𝑛

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

n xi yi xi.yi1 2.5 5 6.25 15.625 39.0625 12.5 31.252 3.5 3.4 12.25 42.875 150.0625 11.9 41.653 5 2 25 125 625 10 504 6 1.6 36 216 1296 9.6 57.65 7.5 1.2 56.25 421.875 3164.063 9 67.56 10 0.8 100 1000 10000 8 807 12.5 0.6 156.25 1953.125 24414.06 7.5 93.758 15 0.4 225 3375 50625 6 90

9 17.5 0.3 306.25 5359.375 93789.06 5.25 91.875

10 20 0.3 400 8000 160000 6 120

∑ 99.5 15.6 1323.25 20508.88 344102 85.75 723.63

10 99.5 1323.25 15.6

99.5 1323.25 20508.88 = 85.75

1323.25 20508.875 344102.31 723.625

metode eliminasi gauss jordan

1 9.95 132.33 1.56

0 333.225 7342.54 = -69.47

0 7342.5375 169003.2563 -1340.645

1 0 -86.92100 3.63435

0 1 22.03477 = -0.20848

0 0 7212.103309 190.11074

1 0 0 5.92559

0 1 0 = -0.78931

0 0 1 0.02636

maka diperoleh :

= 5.92559

= -0.78931

= 0.02636

persamaan kurva diperoleh :

xi2 xi3 xi4 xi2.yi

a0

a1

a2

a0

a1

a2

a0

a1

a2

a0

a1

a2

a0

a1

a2

y=

Maka Kurva dari soal diatas dapat di gambarkan sebagai berikut :

y=

x y

-3 8.53076

-2 7.60965-1 6.741260 5.92559

1 5.162642 4.452413 3.79494 3.190115 2.638046 2.138697 1.692068 1.298159 0.95696

10 0.6684911 0.4327412 0.2497113 0.119414 0.0418115 0.0169416 0.0447917 0.1253618 0.2586519 0.4446620 0.6833921 0.9748422 1.3190123 1.715924 2.16551

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

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

-5 0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

Polynominal a0 + a1x1 + a2x2

Datay= a0 + a1x + a2x2

x

y

-5 0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

Polynominal a0 + a1x1 + a2x2

Datay= a0 + a1x + a2x2

x

y

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

n xi yi xi.yi1 2.5 5 6.25 15.625 39.0625 97.65625 244.140625 12.52 3.5 3.4 12.25 42.875 150.0625 525.2188 1838.26563 11.93 5 2 25 125 625 3125 15625 104 6 1.6 36 216 1296 7776 46656 9.65 7.5 1.2 56.25 421.875 3164.063 23730.47 177978.516 96 10 0.8 100 1000 10000 100000 1000000 87 12.5 0.6 156.25 1953.125 24414.06 305175.8 3814697.27 7.58 15 0.4 225 3375 50625 759375 11390625 6

9 17.5 0.3 306.25 5359.375 93789.06 1641309 28722900 5.25

10 20 0.3 400 8000 160000 3200000 64000000 6

∑ 99.5 15.6 1323.25 20508.88 344102 6041114 109170565 85.75

10 99.5 1323.25 20508.88 15.6

99.5 1323.25 20508.88 344102.31 = 85.75

1323.25 20508.875 344102.31 6041113.72 723.625

20508.875 344102.313 6041113.72 109170564.6 8655.4375

metode eliminasi gauss jordan

1 9.950 132.32500 2050.888 1.56

0 333.225 7342.53750 140039.006 = -69.47

0 7342.538 169003.26 3327276.834 -1340.645

0 140039.006 3327276.83 67109169.202 -23338.408

1 0 -86.92100 -2130.63583 3.63435366

0 1 22.03477 420.25360 = -0.2084778

0 0 7212.10331 241549.00827 190.110736

0 0 241549.01 8257272.5207 5856.61017

1 0 0 780.53727 5.9255875

0 1 0 -317.73884 = -0.7893135

0 0 1 33.49217 0.02635996

0 0 0 167271.463 -510.61139

1 0 0 0 8.30824871

0 1 0 0 = -1.7592403

0 0 1 0 0.12859787

xi2 xi3 xi4 xi5 xi6

a0

a1

a2

a3

a0

a1

a2

a3

a0

a1

a2

a3

a0

a1

a2

a3

a0

a1

a2

0 0 0 1 -0.0030526

maka diperoleh :

= 8.30825

= -1.75924

= 0.12860

= -0.00305

persamaan kurva diperoleh :

y=

Maka Kurva dari soal diatas dapat di gambarkan sebagai berikut :

x y-3 14.82572-2 12.36553-1 10.199140 8.308251 6.674562 5.279773 4.105584 3.133695 2.34586 1.723617 1.248828 0.903139 0.66824

10 0.5258511 0.4576612 0.4453713 0.4706814 0.5152915 0.560916 0.5892117 0.5819218 0.5207319 0.3873420 0.1634521 -0.1692422 -0.6290323 -1.23422

a3

a0

a1

a2

a3

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

-5 0 5 10 15 20 25

-4

-2

0

2

4

6

8

10

12

14

16

Polynominal a0 + a1x1 + a2x2 + a3x3

Data

y=a0 + a1x1 + a2x2 + a3x3

x

y

31.25 78.12541.65 145.775

50 25057.6 345.667.5 506.2580 800

93.75 1171.87590 1350

91.875 1607.813

120 2400

723.625 8655.438

xi2.yi xi3.yi

-5 0 5 10 15 20 25

-4

-2

0

2

4

6

8

10

12

14

16

Polynominal a0 + a1x1 + a2x2 + a3x3

Data

y=a0 + a1x1 + a2x2 + a3x3

x

y

TUGAS 3 METODE NUMERIK DAN KOMPUTASI

NAMA: ZELVIA ZAHARA RNPM: 148110018

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)= 0.000218-6649.78842100742 + 0.000293

-13453.6090026603 +76787.3944372625 -29261.1519561905

0.00044414666.9575779363 + 0.000521

6974.91960988839 +14453.5417531652 -13869.5038378549

0.0005924776.6911127344641553.3092040302

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)