Notes on important topics.

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)
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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
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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):-
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4. Take values of x=1,3,4,5,7 Take y = 7,6,8,9,10 Find
Lagrange’s Polynomial
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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

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By solving the equation we get, Lagrange’s polynomial:
𝒚 = 𝟎. 𝟎𝟔𝟗𝟑𝒙𝟒
− 𝟏. 𝟐𝟑𝟔𝒙𝟑
+ 𝟕. 𝟓𝟗𝟕𝟏𝒙𝟐
− 𝟒𝟐. 𝟏𝟕𝟑𝟕𝒙 + 𝟏𝟖. 𝟏𝟔𝟔𝟓