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Topic:Explain interpolation. Take values of x=1,3,4,5,7
Take y = 7,6,8,9,10 Find Lagrange’s Polynomial
Name: SREEJA SETH
University Roll No:11700220085
Class Roll No: IT2020/005
University Registration No:201170100210019
Paper Name: Numerical Methods
Paper Code: OEC-IT601
Continuous Assessment – 1 (Academic Session: 2023 – 2024)
Department of Information Technology 1
Interpolation
• Interpolation is a useful Mathematical and Statistical tool that is used to estimate values between
any two given points.
• Interpolation is a process of determining the unknown values that lie in between the known data
points.
• It is mostly used to predict the unknown values for any geographical related data points such as
noise level, rainfall, elevation etc.
Types of Interpolation:
 Lagrange’s Interpolation
 Newton’s backword Interpolation
 Newton’s forward Interpolation
 Newton’s divided difference Interpolation
Department of Information Technology 2
Lagrange’s Interpolation
If the distance between the x values are not equal then we have to use Lagrange’s Interpolation.
The value of f(x) ,using Lagrange’s Interpolation is given by from the given table(table:1):-
Department of Information Technology 3
Take values of x=1,3,4,5,7 Take y = 7,6,8,9,10 Find
Lagrange’s Polynomial
Department of Information Technology 4
Y=
𝑥−3 𝑥−4 𝑥−5 𝑥−7 ×7
(1−3)(1−4)(1−5)(1−7)
+
𝑥−1 𝑥−4 𝑥−5 𝑥−7 ×6
(3−1)(3−4)(3−5)(3−7)
+
𝑥−1 𝑥−3 𝑥−5 𝑥−7 ×8
(4−1)(4−3)(4−5)(4−7)
+
𝑥−1 𝑥−3 𝑥−4 𝑥−7 ×9
(5−1)(5−3)(5−4)(5−7)
+
𝑥−1 𝑥−3 𝑥−4 𝑥−5 ×10
(7−1)(7−3)(7−4)(7−5)
𝑦 =
𝑥−3 𝑥−4 𝑥−5 𝑥−7 ×7
144
+
(𝑥−1) 𝑥−4 𝑥−5 𝑥−7 ×6
(−16)
+
𝑥−1 𝑥−3 𝑥−5 𝑥−7 ×8
9
+
𝑥−1 𝑥−3 𝑥−4 𝑥−7 ×9
(−16)
+
𝑥−1 𝑥−3 𝑥−4 𝑥−5 ×10
144
Department of Information Technology 5
By solving the equation we get, Lagrange’s polynomial:
𝒚 = 𝟎. 𝟎𝟔𝟗𝟑𝒙𝟒
− 𝟏. 𝟐𝟑𝟔𝒙𝟑
+ 𝟕. 𝟓𝟗𝟕𝟏𝒙𝟐
− 𝟒𝟐. 𝟏𝟕𝟑𝟕𝒙 + 𝟏𝟖. 𝟏𝟔𝟔𝟓

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11700220085_IT2020005.pptx

  • 1. Topic:Explain interpolation. Take values of x=1,3,4,5,7 Take y = 7,6,8,9,10 Find Lagrange’s Polynomial Name: SREEJA SETH University Roll No:11700220085 Class Roll No: IT2020/005 University Registration No:201170100210019 Paper Name: Numerical Methods Paper Code: OEC-IT601 Continuous Assessment – 1 (Academic Session: 2023 – 2024) Department of Information Technology 1
  • 2. Interpolation • Interpolation is a useful Mathematical and Statistical tool that is used to estimate values between any two given points. • Interpolation is a process of determining the unknown values that lie in between the known data points. • It is mostly used to predict the unknown values for any geographical related data points such as noise level, rainfall, elevation etc. Types of Interpolation:  Lagrange’s Interpolation  Newton’s backword Interpolation  Newton’s forward Interpolation  Newton’s divided difference Interpolation Department of Information Technology 2
  • 3. Lagrange’s Interpolation If the distance between the x values are not equal then we have to use Lagrange’s Interpolation. The value of f(x) ,using Lagrange’s Interpolation is given by from the given table(table:1):- Department of Information Technology 3
  • 4. Take values of x=1,3,4,5,7 Take y = 7,6,8,9,10 Find Lagrange’s Polynomial Department of Information Technology 4 Y= 𝑥−3 𝑥−4 𝑥−5 𝑥−7 ×7 (1−3)(1−4)(1−5)(1−7) + 𝑥−1 𝑥−4 𝑥−5 𝑥−7 ×6 (3−1)(3−4)(3−5)(3−7) + 𝑥−1 𝑥−3 𝑥−5 𝑥−7 ×8 (4−1)(4−3)(4−5)(4−7) + 𝑥−1 𝑥−3 𝑥−4 𝑥−7 ×9 (5−1)(5−3)(5−4)(5−7) + 𝑥−1 𝑥−3 𝑥−4 𝑥−5 ×10 (7−1)(7−3)(7−4)(7−5) 𝑦 = 𝑥−3 𝑥−4 𝑥−5 𝑥−7 ×7 144 + (𝑥−1) 𝑥−4 𝑥−5 𝑥−7 ×6 (−16) + 𝑥−1 𝑥−3 𝑥−5 𝑥−7 ×8 9 + 𝑥−1 𝑥−3 𝑥−4 𝑥−7 ×9 (−16) + 𝑥−1 𝑥−3 𝑥−4 𝑥−5 ×10 144
  • 5. Department of Information Technology 5 By solving the equation we get, Lagrange’s polynomial: 𝒚 = 𝟎. 𝟎𝟔𝟗𝟑𝒙𝟒 − 𝟏. 𝟐𝟑𝟔𝒙𝟑 + 𝟕. 𝟓𝟗𝟕𝟏𝒙𝟐 − 𝟒𝟐. 𝟏𝟕𝟑𝟕𝒙 + 𝟏𝟖. 𝟏𝟔𝟔𝟓