1. The document discusses interpolation methods, including linear interpolation and Lagrange interpolation. 2. Lagrange interpolation approximates a function that passes through given data points, where the degree of the Lagrange polynomial equals the number of points minus one. 3. For two given points, the linear Lagrange interpolation formula estimates values between the points. An example calculates the y-value at x=8 using two known points.

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Numerical Analysis
Lecture 5

2. Interpolation
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• Interpolation enables us to estimate the data points in between the
given data.
𝑦 = 𝑦1 +
𝑦2 − 𝑦1
𝑥2 − 𝑥1
× 𝑥 − 𝑥1
• Linear Interpolation Formula:

3. Lagrange Interpolation
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• The Lagrange interpolation is a method used to approximate the
function of the line or curve that passes through a set of points.
• The degree of Lagrange interpolating polynomial is determined based
on the number of known points
degree = 𝑛 − 1
where n is the number of known points.

4. Linear Lagrange Interpolation
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Given two points 𝑥1, 𝑦1 and 𝑥2, 𝑦2 ,
the linear Lagrange interpolating formula is given by:
𝑦 =
𝑥 − 𝑥2
𝑥1 − 𝑥2
𝑦1 +
𝑥 − 𝑥1
𝑥2 − 𝑥1
𝑦2

5. Linear Lagrange Interpolation
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Example
Determine the value of y at x = 8 given some set of values (2,6), (5,9) by
using the Lagrange interpolation formula.
Solution
Since we have two known points (𝑛 = 2), then we will use the linear
Lagrange formula.
𝑦 =
(𝑥 − 𝑥2)
(𝑥1 − 𝑥2)
𝑦1 +
(𝑥 − 𝑥1)
(𝑥2 − 𝑥1)
𝑦2

6. Linear Lagrange Interpolation
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𝑦 =
(𝑥 − 𝑥2)
(𝑥1 − 𝑥2)
𝑦1 +
(𝑥 − 𝑥1)
(𝑥2 − 𝑥1)
𝑦2
Given, 𝑥1 = 2, 𝑦1 = 6, 𝑥2 = 5, 𝑦2 = 9
The value of y at x = 8 will be:
𝑦 =
(8 − 5)
(2 − 5)
× 6 +
(8 − 2)
(5 − 2)
× 9 =
3
−3
× 6 +
6
3
× 9 = −6 + 18 = 12

7. Second degree Lagrange interpolation
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Given three points 𝑥1, 𝑦1 , 𝑥2, 𝑦2 and 𝑥3, 𝑦3
the second degree Lagrange interpolating formula that approximates
the curve that passes through the three points is given by:
𝑦 =
𝑥 − 𝑥2 𝑥 − 𝑥3
𝑥1 − 𝑥2 𝑥1 − 𝑥3
𝑦1 +
𝑥 − 𝑥1 𝑥 − 𝑥3
𝑥2 − 𝑥1 𝑥2 − 𝑥3
𝑦2 +
𝑥 − 𝑥1 𝑥 − 𝑥2
𝑥3 − 𝑥1 𝑥3 − 𝑥2
𝑦3

8. Second degree Lagrange interpolation
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Example
Given the three data points (2, 0.5), (2.75, 0.36), and (4, 0.25), find the
second Lagrange interpolating polynomial for f (3).

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