Unit 4: Interpolation and Extrapolation

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Unit 4: Interpolation and Extrapolation
Meaning
Let x0, x 1 , x 2 … x n be given values of X and y 0, y1, y2 … yn be the values of Y. If we want to estimate Y for
any value of X between x 0 and x n it can be done by interpolation
But if we want to estimate Y for any value of X outside the range of the given X series, we use the
technique of Extrapolation
Assumptions of Interpolation and Extrapolation
1. No sudden jumps in values of dependent variable from one period to another.
2. There is a sort of uniformity in rise or fall of values of dependent variable.
3. No consecutive missing values in series.
Methods of Interpolation and Extrapolation
Binomial Expansion Method
It is applicable if:
1) Values of the independent variable should have a common difference. That is the values should
be in arithmetic progression.
2) The value of X for which the value of Y is to be interpolated must be one of the values of X
Newton’s Advancing Difference Method
The formula is
๐‘ฆ๐‘ฅ = ๐‘ฆ0 + ๐‘ฅโˆ†10 +
๐‘ฅ=
๐‘ฅ(๐‘ฅ − 1)
2!
2
0
+
๐‘ฅ(๐‘ฅ − 1)(๐‘ฅ − 2)
3!
3
0+
๐‘ฅ(๐‘ฅ − 1)(๐‘ฅ − 2)(๐‘ฅ − 3)
4!
๐‘กโ„Ž๐‘’ ๐‘ฃ๐‘Ž๐‘™๐‘ข๐‘’ ๐‘œ๐‘“ ๐‘‹ ๐‘ก๐‘œ ๐‘๐‘’ ๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘œ๐‘™๐‘Ž๐‘ก๐‘’๐‘‘ − ๐‘ฃ๐‘Ž๐‘™๐‘ข๐‘’ ๐‘œ๐‘“ ๐‘‹ ๐‘Ž๐‘ก ๐‘œ๐‘Ÿ๐‘–๐‘”๐‘–๐‘›
๐ท๐‘–๐‘“๐‘“๐‘’๐‘Ÿ๐‘’๐‘›๐‘’ ๐‘๐‘’๐‘ก๐‘ค๐‘’๐‘’๐‘› ๐‘ก๐‘ค๐‘œ ๐‘Ž๐‘‘๐‘—๐‘œ๐‘–๐‘›๐‘–๐‘›๐‘” ๐‘ฃ๐‘Ž๐‘™๐‘ข๐‘’๐‘  ๐‘œ๐‘“ ๐‘‹
๐‘ฅ=
๐‘กโ„Ž๐‘’ ๐‘ฆ๐‘’๐‘Ž๐‘Ÿ ๐‘œ๐‘“ ๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘œ๐‘™๐‘Ž๐‘ก๐‘–๐‘œ๐‘› − ๐‘กโ„Ž๐‘’ ๐‘ฆ๐‘’๐‘Ž๐‘Ÿ ๐‘œ๐‘“ ๐‘œ๐‘Ÿ๐‘–๐‘”๐‘–๐‘›
๐ท๐‘–๐‘“๐‘“๐‘’๐‘Ÿ๐‘’๐‘›๐‘๐‘’ ๐‘๐‘’๐‘ก๐‘ค๐‘’๐‘’๐‘› ๐‘ก๐‘ค๐‘œ ๐‘Ž๐‘‘๐‘—๐‘œ๐‘–๐‘›๐‘–๐‘›๐‘” ๐‘ฆ๐‘’๐‘Ž๐‘Ÿ๐‘ 
Conditions for using Newton’s Formula is that it must be used when the value to be interpolated is in
the beginning of the data.
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