metode numerik tkm4104 interpolasi -...
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METODE NUMERIKTKM4104
INTERPOLASI
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ENGINEERING DATA
• Tabulasi dan Discrete
• Contoh Data
x y
0.2 10.1
0.3 12.5
0.4 14.2
0.5 17.8
0.6 19.3
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ENGINEERING DATA
• Manipulasi data discrete Interpretrasi
1. Numerical Interpolation.
2. Curve Fitting.
3. Numerical Differentiation.
4. Numerical Integration.
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ENGINEERING DATA
• Perbedaan antara regresi dan interpolasi
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INTERPOLASI
• Mencari data yang berada dalam range data discrete
• Menggambar kurva (jika hanya diketahui titik-titik discrete saja)
• Ketelitian data sangat tinggi
• Terdiri dari:• Interpolasi Polinom
• Interpolasi Lagrange
• Interpolasi Newton
• Interpolasi Newton-Gregory
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INTERPOLASI POLINOM
• Secara umum, penentuan polinom interpolasi kurang disukai, karena sistem persamaan lanjar yang diperoleh adakemungkinan berkondisi buruk, terutama untuk derajat polinom yang semakin tinggi.
• Terdiri dari:
• Interpolasi Linier
• Interpolasi Kuadratik
• Interpolasi Kubik
• Dst
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INTERPOLASI LINIER
• Interpolasi paling sederhana
• Menganggap hubungan berupa garis lurusantara dua titik data
• Persamaan garis lurus yang menghubungkandua titik data tersebut :
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INTERPOLASI LINIER
Untuk contoh data di
atas misalnya ingin
dicari untuk x = 0,25y = 10,1 + ((12,5-0,1)/(0,3-
0,2))*(0,25 – 0,2)
= 11,3
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INTERPOLASI KUADRATIK
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INTERPOLASI KUADRATIK
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INTERPOLASI KUADRATIK
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INTERPOLASI KUBIK
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INTERPOLASI KUBIK
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INTERPOLASI LAGRANGE
Tinjau Interpolasi Linier
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INTERPOLASI LAGRANGE
Bentuk umum interpolasi Lagrange
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INTERPOLASI LAGRANGE
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INTERPOLASI LAGRANGE
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INTERPOLASI NEWTON
Tahapan pembentukan interpolasi Newton
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INTERPOLASI NEWTON
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INTERPOLASI NEWTON
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INTERPOLASI NEWTON
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INTERPOLASI NEWTON
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INTERPOLASI NEWTON-GREGORY
• Tabel selisih
• Ada dua macam tabel selisih, yaitu tabel selisihmaju (forward difference) dan tabel selisihmundur (backward difference)
• Karena itu, ada dua macam interpolasiNewton-Gregory, yaitu Newton-Gregory majudan Newton-Gregory mundur.
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INTERPOLASI NEWTON-GREGORY MAJU
Bentuk interpolasi polinomialnya adalah :
Pn(x) = C0 + C1 (x - x0) + C2 (x - x0) (x - x1) +…. + Cn (x - x0) (x - x1) … (x - xn-1)
dimana :
C0, C1, … , Cn suatu konstanta Cj ; j = 0, 1, … , n
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INTERPOLASI NEWTON-GREGORY MAJU
Pn(x) = yi ; i = 0, 1, 2, … , n
P0(x0) = f(x0) = y0
C0 = y0
P1(x1) = f(x1) = y1
C0 + C1(x1 - x0) = y1
C0 + C1(x1 - x0) + C2 (x2 - x0)(x2 - x1) = y1
C0 + C1(xn - x0)+…+ Cn (xn - x0)(xn - x1)…(xn - xn-1)= yn
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INTERPOLASI NEWTON-GREGORY MAJU
Harga Cj dapat dirumuskan sebagai:
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INTERPOLASI NEWTON-GREGORY MAJU
Harga Cj dapat sederhanakan menjadi:
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INTERPOLASI NEWTON-GREGORY MAJU
Harga Cj dapat sederhanakan menjadi:
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INTERPOLASI NEWTON-GREGORY MAJU
Secara umum harga Cj dapat disimpulkanmenjadi:
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INTERPOLASI NEWTON-GREGORY MAJU
Tabel selisih
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INTERPOLASI NEWTON-GREGORY MAJU
Contoh
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INTERPOLASI NEWTON-GREGORY MAJU
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INTERPOLASI NEWTON-GREGORY MAJU
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INTERPOLASI NEWTON-GREGORY MAJU
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KERJAKAN!!!
Derajat 3 = Third Order