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

, y

1

, y

2

… y n

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‘ฆ ๐‘ฅ

= ๐‘ฆ

0

+ ๐‘ฅโˆ†

1

0

+ 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‘ฅ = 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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