# Unit 4: Interpolation and Extrapolation

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```Unit 4: Interpolation and Extrapolation
Binomial Expansion
The steps are as follows:
1. Find the number of known values (n) of dependent variable Y.
2. For the two missing values, first take
βπ0 = 0 π‘βππ‘ ππ  (π¦ − 1) π = 0
And β1π = 0
is written by increasing the suffix value by one in
βπ0 = 0
3. The left hand side of the equation is expanded using the binomial expansion. Suppose 4 values
of Y are known, 4th order (leading) difference will be zero.
β40 = 0
(π¦ − 1) 4 = π¦4 − 4π¦3 + 6π¦2 − 4π¦2 + π¦0 = 0
We get one missing value within the range of the data.
4. To get the other missing value, the same binomial expansion as in case of
β40 = 0 Is written with suffixes of each ‘raised by 1.
π¦5 − 4π¦4 + 6π¦3 − 4π¦2 + π¦1 = 0
The successive numerical co-efficient of ‘y’ can be obtained using binomial coefficients table or y the
formula
πβπ πΆπππππππππ‘ ππ ππππ£πππ’π  π¦ &times; π π’ππππ₯ ππ ππππ£πππ’π  π¦
ππππ’πππ‘πππ πππππ ππ ππππ£πππ’π  π‘πππ
The numerical coefficient of the first term will be always 1.
Newton’s Advancing Difference Method
π¦π₯ = π¦0 + π₯β10 +
π₯=
π₯(π₯ − 1)
2!
2
0
+
π₯(π₯ − 1)(π₯ − 2)
3!
3
0+
π₯(π₯ − 1)(π₯ − 2)(π₯ − 3)
4!
π‘βπ π£πππ’π ππ π π‘π ππ πππ‘πππππππ‘ππ − π£πππ’π ππ π ππ‘ ππππππ
π·ππππππππ πππ‘π€πππ π‘π€π πππππππππ π£πππ’ππ  ππ π
π₯=
π‘βπ π¦πππ ππ πππ‘πππππππ‘πππ − π‘βπ π¦πππ ππ ππππππ
π·πππππππππ πππ‘π€πππ π‘π€π πππππππππ π¦ππππ
```