INTERPOLASI (METODE NUMERIK)

Erik Pebrinasyah

Ali Martun Pulungan

Nyella Kenanga Andini

Dwi Yulia Ningsih

Ario Amanda

Tentukakanan lah 5 titik Interpolasi (sebarang titik) dari data berikut dengan mengugunakan 4 metode yaitu

1. Interpolasi Liner

2. Interpolasi kuadrat

3. Interpolasi Newton

4. Interpolasi Lagrange

(Sumber: Nasution;54)

x y

15 14944

20 19867

25 24740

30 29552

35 34290

40 38945

45 43497

50 47943

55 52269

60 56464

Penyelesaian :

Misalkan diambil sembarang lima titik interpolasi yaitu 18, 23,33,43, dan 53.

1. Interpolasi Linier

Untuk x = 18

X0 = 15 → f(x0) = 14944

X1 = 20 → f(x1) = 19867

Untuk x = 23

X0 = 20 → f(x0) = 19867

X1 = 25 → f(x1) = 24740

Untuk x = 33

X0 = 30 → f(x0) = 29552

X1 = 35 → f(x1) = 34290

Untuk x = 43

X0 = 40 → f(x0) = 38945

X1 = 45 → f(x1) = 43497

Untuk x = 53

X0 = 50→ f(x0) = 47934

X1 = 55 → f(x1) = 52269

Tabel Interpolasi linier

2. Interpolasi Kuadrat

Untuk x = 18

X0 = 15 → f(x0) = 14944

X1 = 20 → f(x1) = 19867

X2 = 25 → f(x2) = 24740

b0 = f(x0)

= 14944

b1 = = 984,6

b2 = = = - 1

F (x) = b0 + b1 ( x – x0) + b2 (x-x0)(x-x1)

= 14944 + 984,6(18-15) + (-1)(18-15)(18-20)

= 14944 + 984,6 (3) + (-1)(3)(-2)

= 14944 + 2953,8 + 6

= 17903,8

Untuk x = 23

X0 = 20 ,f(x0 ) =19867

X1 = 25 ,f(x1 ) =24740

X2 = 30 ,f(x2 ) =29552

b0 = f(x0 )

= 19867

b1 = = = 974,6

b2 = = = = -1,22

F (x) = b0 + b1(x- x0) + b2(x- x0) (x- x1)= 19867 + 974.6 (23-20) + (-1.22)(23-20)(23-25)= 19867 + 2923.8 + 7.32= 22798.12

Untuk x = 33 X0 = 30 , f(x0 ) =29552 X1 = 35 ,f(x1 ) =34290 X2 = 40 ,f(x2 ) =38945

b0 = f(x0 )= 29552

b1 = = 947,6

b2 = = = -1,66

F (x) = b0 + b1(x- x0) + b2(x- x0) (x- x1)

= 29552 + 947.6 (33-30) + (-1.66)(33-30)(33-35)

= 29552 + 2842.8 + 9.96

= 32404.76

untuk x = 43

X0 = 40 → f(x0) = 38945

X1 = 45 → f(x1) = 43497

X2 = 50 → f(x2) = 47943

b0 = f(x0)

= 38945

b1 = = = 910,4

b2 = = = -2,12

F (x) = b0 + b1 ( x – x0) + b2 (x-x0)(x-x1)

= 38945 + 910,4(43-40) + (-2,12)(43-40)(43-45)

= 38945 + 910,4 (3) + (-2,12)(3)(-2)

= 38945 + 2731,2 + 12,72

= 41688,92

Untuk x =53

X0 = 50 → f(x0) = 47943

X1 = 55 → f(x1) = 52269

X2 = 60 → f(x2) = 56464

b0 = f(x0)

= 47943

b1 = = 865,2

b2 = = -13,1

F (x) = b0 + b1 ( x – x0) + b2 (x-x0)(x-x1)

= 47943 + 865,2(53-50) + (-13,1)(53-50)(53-55)

= 47943 + 2595,6 + 78,6

= 50617,2

Tabel Interpolasi Kuadrat

3. Interpolasi newton.

Tabel Hasil Interpolasi Newton

4. Interpolasi Lagrange

Tabel Hasil Interpolasi Lagrange